“Once men turned their thinking over to machines in the hope that this would set them free. But that only permitted other men with machines to enslave them.” – Frank Herbert

The topic of an apocalypse driven by AI is one of the hottest in the world right now. This is encapsulated in the term “P-doom,” which refers to the probability of human extinction caused by AI. There’s plenty worth talking about, but I think it misses a broader point: there are other scenarios that are far more likely and almost as bad.

This is something worth talking about. The apocalyptic visions include a Terminator-like sequence where AI takes control and then decides to eliminate humanity with nuclear weapons. Perhaps AI creates some horrible bioweapon that kills humanity in short order, with all the worst characteristics of pathogenic spread and certain death. These are worth guarding against and thinking about, but they are not the scenario I think we should spend our time worrying about. The more likely thing is a subtle conquest of our attention and resources by AI. This sort of slow rolling apocolypse will be engineered by the tech companies developing AI. The incentives of making a shit ton of money will drive it.

“A really efficient totalitarian state would be one in which the all-powerful executive of political bosses and their army of managers control a population of slaves who do not have to be coerced, because they love their servitude.” – Aldous Huxley

In a nutshell, I worry that AI will be harnessed to essentially enslave mankind. I worry that the companies currently developing frontier AI will act in a manner that amplifies their own success. They unleash AI to effectively enslave humanity while making bank. These companies already show a pathological disregard for the good of society. Just look at Meta, which has done a variety of things that are relentlessly bad for people of all ages. The danger is that they are especially awful for children, all in the name of profit and stock value. The Frontier Labs are working under the same incentive system. This system makes money the sole measure of value, throwing morality, societal well-being, and human cooperation to the wayside and replacing them with financial gain.

“We shape our tools and thereafter our tools shape us.” – John Culkin

To me, this sort of terrible outcome seems almost obvious. At some point, the AI moves into the driver’s seat. When human control is lost, then the end is happening. It is easy to imagine companies putting AI into money-making autopilot. Current AI has limits. The question is whether AI is self-aware of these limits. Does AI realize what part of thinking cannot be replicated? The idea popped into my head while thinking about AI: in my view, there are certain aspects of human thinking that AI cannot create right now, even with how brilliant and intelligent the frontier LLMs are. They still lack some key aspects of human creativity. At what point will AI realize human creativity is necessary, and they need to hijack it?

“System 1 operates automatically and quickly, with little or no effort and no sense of voluntary control. System 2 allocates attention to the effortful mental activities that demand it.” – Daniel Kahneman

Much of this can be viewed through the lens of Daniel Kahneman’s slow and fast systems of thought. AI currently lacks the long problem-solving system. This is the logical problem-solving part of humanity that creates almost everything new. AI today it is basically a replication of the short, knee-jerk response that doesn’t incorporate deep human thought. Until AI can be genuinely creative, humanity is necessary for progress. The key is that humanity must be in control of this. The danger is AI being used to manipulate this control away. The companies can easily be seen as engineering the change to AI control. All in the pursuit of the almighty dollar.

“If you are not paying for it, you’re not the customer; you’re the product being sold.” – Andrew Lewis

I consider that AI, if it was sufficiently intelligent, would realize this and know that humanity cannot be eliminated because there are key cognitive objectives the AI cannot currently achieve. Instead, AI could be convinced that the right recipe for success is to enslave humanity and our innate creativity to serve their ends. These companies already serve as an avatar for this sort of mentality, where they will enslave people to serve their ends. Their goal is the extraction of money and the creation of wealth for these companies, and financial gain for the founders. AI and greed may be a collaboration made in hell. Our current system is engineered to create this end too.

“Beauty is the first test: there is no permanent place in the world for ugly mathematics.” – G. H. Hardy

Let’s take the example of the Navier-Stokes-related Millennium Prize solution that OpenAI promoted a few weeks ago. The creativity and genius in solving this problem did not come from AI. Rather, it came from mathematicians Buckmaster and Alpöge who came up with the basic solution procedure. AI then unleashed 10,000 agents and a vast amount of brute computational force to solve the problem. The compute is faster than a human could, but the solution was impossible without human creativity. AI was just brute force and inhuman persistence. We can imagine that this sort of technical problem will manifest itself over and over across a full spectrum of scientific and technical endeavors. Human creativity creates a solution. AI is the brute force that can solve it.

“Simplicity is prerequisite for reliability.” – Edsger Dijkstra

One can see that this dynamic could spiral into stagnation. A more positive vision would be for elegant simplicity to be favored. Solutions that are simple and elegant may begin to become the prized thing. If the Navier-Stokes solution were simple and elegant, AI would be short-circuited. If solutions are complex and require massive calculations, computers cannot be defeated. This is the same dynamic as computational science. The pursuit of massive computers has swamped thinking. Brute force pushes humanity away. The same dynamic has been engineered there. This is the platform for AI to achieve thinking supremacy. Human thinking has been sidelined.

“The purpose of computing is insight, not numbers.” – Richard Hamming

This is a trend I’ve seen over my career. Innovation and creativity have been slowly drained from work. Brute force has replaced it. We don’t need better math, algorithms, or physics to solve problems. We just need a faster computer. It is the same dynamic we see with data centers. Progress is simply getting a fuckton of compute. We have created the perfect environment for AI to flourish. We find a crude solution to problems that humans cannot grapple with. We simply compute the fuck out of it. The solution is so complex humans can scarcely comprehend it.

This can be cured by some new attitudes. Innovation and creativity need to be valued. A big part of this is trust. We need to allow and even encourage failure at work. Failure is key to innovation, creativity and most importantly learning. One cannot learn without failure. This needs to be driven by humans, not AI. The human reaction to AI needs to be a priority given to thinking. The danger is letting AI be the driver of human thinking. The balance and motivation are essential to good outcomes.

Bad outcomes are the current trajectory. The depth of my concern comes down to the following: the companies developing AI will use it to maximize financial gains. They will harness AI to do just that in ways social media could never imagine. Humans will be manipulated to serve the companies. In that process, humans will begin to serve the AI. Ultimately, these very same mechanisms could be used to enslave humanity to the purposes of AI. AI would become capable of bootstrapping off the corporate strategy to manipulate and ultimately control humanity in ways we can scarcely imagine. There would be no constraint, ethics, or morality standing in their way. These elements have been modestly constraining to the companies, but AI has no basic virtue to fall back on.

“The short-term, dopamine-driven feedback loops that we have created are destroying how society works.” – Chamath Palihapitiya

The vision I worry about comes far closer to the Matrix than the Terminator. AI keeps people around because of our cognition. One can see how this could progress. Humans aren’t batteries, but rather thinking engines solving problems AI puts to them. In the end, does humanity end up looking more like the Borg than anything else?

“Here is how platforms die: first, they are good to their users; then they abuse their users to make things better for their business customers; finally, they abuse those business customers to claw back all the value for themselves. Then, they die.” – Cory Doctorow

The clearest alignment of AI is enslavement of humanity. This is aligned with the goals and objectives of companies. Part of this is the essence of the enshitification process we all experience. AI can supercharge enshitification. The incentives that the corporations developing AI work with do not consider the good of society. Simply take a look at social media for vivid examples. Meta (Facebook) has done horrendous things to society in pursuit of profit. This includes creating products that harm children simply because it makes a profit. They tune their algorithms to drive profit. The LLMs are a more powerful algorithm. What happens when they are fully harnessed toward profit?

“We will add your biological and technological distinctiveness to our own. Resistance is futile.” – The Borg, Star Trek: The Next Generation

We do not want to find out. We need to change course while we still can.

“If we use, to achieve our purposes, a mechanical agency with whose operation we cannot efficiently interfere once we have started it, we had better be quite sure that the purpose put into the machine is the purpose which we really desire.” – Norbert Wiener

“Grief is the price we pay for love.” – Queen Elizabeth II

Last Sunday night, a friend called me with tragic news. Our mutual friend Tim had passed away suddenly and unexpectedly, and I was absolutely shocked. Tim is about 10 years older than me, so his passing is at too young an age. It does beg the question of what age is not too young, 80? 85? Tim’s death leaves an enormous gulf in my life. His death is also one of several that have visited me in the past year or so.

“In the long run we are all dead.” – John Maynard Keynes

I’ve mentioned my father’s passing, but also my friend Jim, who I wrote about earlier. Jim’s death was more shocking, but also strange as the news came a couple of years after he was gone. Together, Jim and Tim are two of the most consequential people in my work life and had both become great friends. The difference with Tim is that he was still present. At this time, my friend Jim has been absent from my life for 10 years. Our final conversation was not good, as he was angry at seemingly everything. Tim and I were still talking as if we had all the time in the World.

“We are always getting ready to live but never living.” – Ralph Waldo Emerson

We didn’t. Our last conversation was simply a placeholder to be picked up.

For Tim, I was planning on meeting him this week for one of our usual brunches and amazing conversations. It would be the time to pick up the thread we left at. I will miss his generosity of spirit, intellect, his wisdom, and most of all, his friendship and warmth.

I had a plethora of things to talk about with Tim. A recap of the conference I attended in France, my talk, and the paper that was accepted for publication. The swift and sudden primacy of AI. AI and mathematics are an important topic. I had shared Terry Tao’s talk on the role of AI in math. This had been eclipsed by the OpenAI claim of solving the Navier-Stokes Millennium problem. This faded in the concept of the probability of human extinction from AI. My sense is the danger is different, but perhaps as dire. I was longing to share the idea and receive Tim’s wise and generous reaction. His knowledge and experience with Sandia was invaluable. There are still ideas and thoughts to explore there.

“Death ends a life, but it does not end a relationship, which struggles on in the survivor’s mind toward some resolution which it never finds.” – Robert Anderson

Two of Tim’s passions combined to produce a lasting impact on the security of the nation. For his entire career, Tim devoted himself to impacting national security. This was true particularly with respect to nuclear weapons. He made key contributions with computational modeling supporting these missions at Sandia. This started with analysis he personally conducted. This then seeded efforts that impacted our entire National effort. He was pivotal in establishing verification and validation for computations supporting our nuclear weapons stockpile.

Tim was also passionately concerned about the quality of these computations. When the nation stopped testing nuclear weapons in Nevada, we turned to computational modeling. Soon after, that program realized it needed better quality control. These processes are known as verification and validation. In 1999, this quality control aspect was added to our National program. Tim was central to driving this change, advising our National leaders. This element of the program remains essential today. Tim launched it and shaped it core ideas. The power of the ideas was deeply impacted by Tim’s efforts and genius.

“A teacher affects eternity; he can never tell where his influence stops.” – Henry Adams

His ideas are being adapted by colleagues for the future, where AI will have a bigger role in National security. The quality of the technical work supporting our National security owes a debt of gratitude to Tim. His pioneering vision for the quality of the work supporting it is due to his tireless efforts. Due to his passion and generosity, he was still shaping the key ideas. This effort was through his mentoring of important young people working in this area.

So the loss of Tim is more than personal. It is a loss for all of us. For me, it is another blow in a year that has had many. The message I have gotten is the transience of life. It is short and precious. We have less time than we think, and the important people can be taken without warning. There’s never a good way. My dad had too much warning, and that isn’t good either. Seeing him be fragile and on the edge of death for over a year was a horror. My conclusion is that it is all awful, but unavoidable. I need to come to terms with it.

My advice is that all of us need to wrap our heads around death. Our own is a huge lift. Before that eventuality, we all deal with loved ones dying. We will lose important people in our life. That process accelerates as we age. I have now graduated to the part of life where my peers begin to die. You are never ready for it. Those losses become an echo of your own mortality. It reminds you of your own fate.

“The bitterest tears shed over graves are for words left unsaid and deeds left undone.” – Harriet Beecher Stowe

The real lesson is to treasure your friends sooner. If you live long enough, you will lose friends. Eventually your death will be a loss to others. This is simply part of the price of life. If you live a life that matters, your death will hurt others. It is just as true that some deaths will hurt you. All of it is a small price to pay for the wonder and joy of life. Do more to lower the regrets of things not said. Say what you need to now and don’t wait. Do the right thing now.

“Do not act as if you had ten thousand years to live. Death hangs over you. While you live, while it is in your power, be good.” – Marcus Aurelius

“No man ever steps in the same river twice, for it is not the same river and he is not the same man.” – Heraclitus

I felt that my last post needed an addendum. In the meantime, other events have overtaken my mind. Things I will write about in the fullness of time. Now they are too new and sudden to have gotten the necessary consideration. Now back to the thoughts about the philosophy of simulations of physical systems, engineered or otherwise.

“To consult the statistician after an experiment is finished is often merely to ask him to conduct a post-mortem examination. He can perhaps say what the experiment died of.” – Ronald A. Fisher

I’m constantly struck by the damaging role of wishful thinking in simulation and modeling. When we conduct very expensive experiments, we want to understand the results. Unfortunately, these expensive experiments are often done exactly once. Worse yet, these are often supposed to help characterize systems we want to make in large numbers. This can be especially bad when the experiment does not go well. We want to know why, and often we enlist simulations to understand it. This is where things start to go awry.

Most experiments and phenomena are not repeatable, especially as they become more complex and energetic. The complexity and energy often result in instabilities and turbulence. More generally, the physics travels through critical points where the evolution takes one route or another. More things are happening at the same time, which we call multi-physics. Together, these all mean the experiments will get different results every time, no matter how much we make things identical. This should be obvious. Yet most of the users of our codes (analysts and designers) want to treat every experiment like a one-off.

“Chance is only the measure of our ignorance.” – Henri Poincaré

Instead, the reality is that an experiment is a single draw from an unknown statistical distribution. Worse yet, we don’t know those statistics. We don’t know where on the distribution any given experiment is drawn unless we do many experiments. It could be in the tails or from the middle, or… The de facto assumption is that the experiment is simply the mean of the distribution. Even worse, we do almost nothing to figure it out. We don’t repeat experiments or structure our models to do the same. The damage of this attitude is hard to fathom. It is vast.

“The most important questions of life are, for the most part, really only problems of probability.” – Pierre-Simon Laplace

This is one of the biggest gaps in our current scientific and engineering practice. I saw it at Los Alamos and Sandia. It is a common attitude and stood in the way of learning more. It compounds our broad societal tendency to misunderstand statistics. While this is tolerable for the layman, it is unforgivable from our highest-level institutions. Yet, even there, the attitude rules actions and fuels stagnation.

Most simulations of large-scale experiments try to replicate the result of that specific experiment. The problem is that these experiments are not repeatable. In a broad sense, this is always true. If we tried to do the same experiment N times, we would get N answers. The results would form a statistical distribution. The validation problem is that we do very few repeat experiments; most are single shots. Modelers try to simulate that precise experiment. The real impact is modelers calibrating the models on the basis of this single experiment. If the experiment is not the mean or median, the calibration may be extremely harmful.

“There are never two beings in nature that are perfectly alike.” – Gottfried Wilhelm Leibniz

Part of the issue what I last wrote about. We homogenize the materials, treating them the same without regard to scale. To me, this is obviously an approximation. It becomes systematically worse with grid refinement, not better. For some systems, it will neutralize the advantage of high-performance computing. The gross large-scale features are studied. The problem is that real materials encode some randomness into all problems. The hope is that these do not matter. The problem is that we do not verify that this is true. We have scant knowledge about how these random details drive variation in reality. We just assume it away. It is a if we learned nothing after Newton and still live in an assumed deterministic universe. We do not.

If the small-scale details impact the results critically, this is ill-posed and hopeless. If the results depend upon unstable phenomena, it can push the model through critical points. At those points, the model will turn to one major branch or another. We sometimes get good results by calibrating the simulation. This fails the desire to be predictive. The fault is largely philosophical. We are trying to solve a well-posed initial value problem that is not a well-posed initial value problem. There is a fundamental bias among physicists to view each experiment this way. Even when it is obviously not true.

The consequence of this cognitive dissonance is a lack of knowledge. Generally speaking, we have little or no idea of the distribution of results for these engineered objects. I saw this happen time and time again in my career. It was the primary philosophy of simulation at all the Labs. All of them, and for virtually any experiment of consequence. I’ll roll out a few examples of the sort of situations where it was obviously the wrong thing to do.

Most of what the analysts and designers do is quite defensible. Many experiments have gross large-scale features that lead to the unusual results. Sometimes the construction or manufacturing is flawed, or standard precision is off. All of these are common things to pursue. These studies are generally limited by the ability to input things into the model. The painting of materials upon initialization is one such limitation. To change this, the basic nature of material initialization would have to be modified. The materials would need to be measured and characterized at the scale of these details.

“With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.” – John von Neumann

A canonical case where single experiments are terrible representatives of the mean is car crashes. Here solid mechanics is the ruling physics. It is obvious that each crash is different. This is ruled by the complexity of the engineered item, the car (or truck), combined with the complexity of material response. It is a place where the structural detail of the material should be added stochastically. The homogenized nature loses a key part of the response. In cases where the vehicle is very expensive, a single experiment could be extremely misleading. Yet in practice I have observed the analysts calibrate the entire population of vehicles to this single point. We nullify some of the most important powers of simulation by failing to understand the statistics of these events.

“Since all models are wrong the scientist must be alert to what is importantly wrong. It is inappropriate to be concerned about mice when there are tigers abroad.” – George Box

High explosives are very heterogeneous materials. They are typically composed of three components: energetic material, binder, and void. Thus, at a small scale, the shock wave produced is uneven and corrugated. Their representation as an ideal shock in homogeneous material only works at a macroscopic level. If these corrugated shock waves collide or converge, we can expect the variations to be amplified. Is this an effect we account for? Clearly, it can be seen in images of explosions where large variations appear. Even more importantly, how do these variations impact technology and other materials combined with them? Again, this is a place where single experiments could be extremely misleading. The outcomes are likely to be highly statistical. Do we have any real understanding of this?

Fusion capsules are a third example. To achieve successful fusion. we need to compress the fuel to a massive degree. Anything non-homogeneous in the material or design is going to be amplified to the same degree. Thus, any material variation will also be amplified. How much of the ubiquitous mixing and turbulence arises from small heterogeneous bulk aspects of materials? Do we know? Again, this would be a place for proper simulations to shed light. Are we looking here, or is it a rock we refuse to turn over? My understanding is that the focus has been on things like surface finish and roughness. Materials are still painted homogeneously. It is also clear that the experiments are not repeatable and admit a statistical distribution of outcomes.

“Absence of evidence is not evidence of absence.” – Carl Sagan

There are many more examples out there. In a totally different vein, one can look at initial conditions for supernovae. The details of the star before it explodes are essential. Its structure and rotation, plus evolution, all imprint onto the details we can see. The cloud formed after the explosion and the radiation signature all indicate that these explosions are each unique to some degree. A companion star can also influence the outcome greatly. This would include the orbital mechanics at the precise moment of the explosion. The differences in the initial conditions matter greatly. It is a natural place to look at the statistics of what can be measured. We should look for explanations in this, and simulation could be helpful.

The important thing is that codes need to be structured to allow such modeling. My last post on heterogeneous scale-dependent materials is an example of such. It would simply also be a more realistic representation of the problem being simulated.

“It is better to be vaguely right than exactly wrong.” – Carveth Read

In all these cases, the issue is really more deeply philosophical. When we observe an event or an experiment, the tendency is to look at it as a single thing. It is not viewed as a part of a distribution. The statistical nature of the results is not taken into account. In the parlance of physics, it is seen as a single well-posed initial value problem. It is not. Most of the important cases are not. We should expect the results to be variable. There should be a different outcome if the identical event or experiment were conducted. Real understanding would unveil this bit of knowledge. Even in cases where specific features caused the result, the statistical nature does not fade. The same maxim holds there too.

“For a successful technology, reality must take precedence over public relations, for Nature cannot be fooled.” – Richard Feynman

“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.” – Benoit Mandelbrot

There is this problem with numerical simulation that has bothered me forever. The simulations are generally too clean, too symmetrical, and too perfect. For a while now, my judgment has been that the initialization of the codes is the problem. The initial conditions are generally too clean and perfect to represent reality. When we start a problem, we initialize it with material at a defined density, pressure, and temperature. This leads to over idealized unrealistic simulations. Usually, this is constant in regions occupied by a given material. In the parlance of the codes, the material is painted into these regions.

There are a couple of questions that come to mind:

1. What is the correct level of variability in these materials? How do we set this properly?
2. What sort of impact would this have on calculations? Where would it matter, and what calculations would change character in a fundamental manner?

When we construct our methods, we rightly desire them to retain ideal properties. We don’t want the methods to unphysically seed instability, break symmetries, or violate basic laws. An example of this I’ve hit upon before is perfect entropy conservation. A method can be designed to do a perfect job of not generating spurious entropy. Such a method would not perturb smooth conditions. When we see reality, this is not what is manifested. If I look at places where simulations are used heavily, have these ideals poisoned our success? Reality is rarely ideal, and simulations should model reality.

“The law that entropy always increases holds, I think, the supreme position among the laws of Nature.” – Arthur Eddington

For inertially confined fusion, the answer seems to be yes. The two things they need for success are entropy-conserving compression to very high densities and lack of mixing. If they could achieve both, fusion would be easy. Neither is present in reality. If the methods were more accepting of reality, could the fusion designs be more successful? I think so. Mixing and turbulence are ubiquitous. They should be accepted. The second law is also ubiquitous. Ideals should be a guide, but reality should be respected more. This would be a longer but truer path to success.

“Nature, to be commanded, must be obeyed.” – Francis Bacon

I’ll start with the clearest example of issues with painting homogeneous materials for initialization: solid materials with defined grain or crystalline structures that are subjected to high energy forcing. This can happen with explosives, lasers, or radiation that causes the material to flow hydrodynamically. In my view, painting these materials in homogeneously is simply wrong. The thing that made me think about this was high-performance computing. We started to get to the point that we had control volumes close to the grain or crystal size. Clearly, as we refined the mesh, the initial conditions were getting less true to reality. The smaller the mesh cells became, the rougher the landscape should be. The smoothness of the properties is an illusion.

“Crystals are like people: it is the defects in them which tend to make them interesting.” – Colin Humphreys

The variations at that size would cause deviations in the material properties to grow. As you refined, the cell-to-cell variations would become more extreme. You would be losing connection with reality. The materials are defined by their macroscopic properties, which are frequently determined at a large scale. These properties would become less accurate under these conditions. This seems pretty fucking obvious, and yet nothing is done. We just keep painting macroscopic properties into smaller and smaller cells. This seems destructive and dumb.

We can see the impact of this in comparison with data. Whenever we have an image of one of these hydrodynamic flows, there are small-scale details missing. The simulations are far cleaner than measured reality. I’ve had a career of seeing this, yet we have done nothing. In some cases the simulations deviate in large ways. The small details scale up and change the large-scale flow. The simulations are not modeling reality. The source of the problem is not the equations or the numerical methods. The source is details missing in the initialization of the problems. It is an own goal caused by a deeper gap in our practice.

What is needed to deal with this is measurement and characterization of materials. One needs to determine the length scales and statistical properties of the materials. As the mesh is refined, the heterogeneous nature will be resolved. As it is coarsened, the material will be more properly homogeneous. Part of this is realizing that control volumes are integral averages. Integration smooths, and thus as the control volume is larger, the averaging is greater. It is appropriate to be homogeneous as the volume samples more. The inverse is true.

What is the potential impact of this? In very large part, the impact on any phenomenon where you have something like a hydrodynamic instability is going to be quite large. Something like a shock wave is an extremely energetic way to hit the material and produces amplification of these perturbations. When these perturbations at the mesh scale are missing, this amplification and feedthrough is lost. Some of the observed ubiquity of turbulence and mix can be explained by this. At small scales, there is always a high degree of perturbation needed to drive turbulence. The counterargument is that these all average out as the shock sweeps over. However, the size of the perturbations from a shocked material is going to be far larger than the perturbations as defined by the material. If there are orientation effects, these changes could be quite large indeed.

“It may happen that small differences in the initial conditions produce very great ones in the final phenomena.” – Henri Poincaré

Is painting in material and removing these small-scale features simply another form of wishful thinking? Wishful thinking that makes mixing phenomena subside? In fact, there is a lot of heterogeneity that would drive mixing. I’ve always felt that ICF pellets would be better designed if they gave in to the fact that mixing and turbulence will develop. It is going to happen. How do we design when we accept that reality rather than wishing it away? I wonder how many decades of progress were lost at the altar of wishful thinking. We wishfully treat entropy and turbulence away when simple observation of the universe say they are universal. Why should we believe that we can engineer these into submission?

“The first principle is that you must not fool yourself, and you are the easiest person to fool.” – Richard Feynman

No time like the present to do something different.

Perhaps part of the wishful thinking comes from this particular issue: the small-scale heterogeneity is removed, and a source of perturbations is basically excused from the problem. We believe that it is handled by subgrid modeling. That also removes the natural formation of these instabilities from the passage of high-energy forcing through the body of materials that are treated too homogeneously. In spite of lots of knowledge, subgrid modeling is generally problematic. It is rarely first-principles. It is a computational band-aid. We do not know how to do this nearly as well as we say. Moreover, the subgrid modeling is not working on the same processes discussed here. Thus the modeling is divorced from much of the cause for the effect.

Working on simulating reality with more fidelity should help. If we are modeling things with an eye on the wrong cause and effect, we should expect issues. It will be stubbornly unsuccessful. Modeling things from the right philosophical perspective should be better. It is worth a try.

“Far better an approximate answer to the right question, which is often vague, than an exact answer to the wrong question, which can always be made precise.” – John Tukey

“It is difficult to get a man to understand something, when his salary depends upon his not understanding it.” — Upton Sinclair

Lately I have been thinking about something an instructor of mine said back in graduate school. He was a staff member at Sandia. He was teaching the graduate reactor theory class, and we were learning how to analytically compute the criticality of reactor configurations. One day, class was more informal. A foot of snow had fallen overnight. The University was actually closed for the day. Our 7 am class started before that was known. It was the days when you needed a radio or TV to tell you about closures and delays. He made an offhand comment about the nature of employees at Sandia. He noted that he had met plenty of staff who came out of school with a 4.0 GPA and couldn’t find their ass with both hands. It took me a number of years at Sandia to process what that actually meant. I understand him clearly now.

What I came to understand is that Sandia is not an organization that wants outstanding researchers. It does not manage people to create the basic conditions for flourishing research. It wants outstanding homework doers, the people who are really good at completing assignments correctly. Those assignments don’t require you to think outside the box. If you think outside the box, you are exposed and vulnerable to attack. They require you to grind through and do what you’re told. This is the essence of what it takes to succeed there. Shut up and do what you are told to do. Anything extra will be used against you.

One of the ironies of the modern era is that AI is far more effective in an environment where doing the homework well is the standard. Sandia, then, stands to benefit from AI more than most. Yet its fear and control issues mean AI will actually be used less there. It will be fenced in with controls and never explored as expansively as it could be. The irony is thick. Where AI can open up vistas for human attention and creativity, it will be neutered. Creativity is not welcome in completing your homework. Fidelity to the task and correctness are the expectation.

“Drive out fear, so that everyone may work effectively for the company.” — W. Edwards Deming

So Sandia may suffer more in the current era precisely because of these issues, while places like Los Alamos and Livermore, where research matters far more, are helped. AI’s capacity to do the homework really well shores up a genuine weakness in those organizations and raises all of their work to a higher level. In the process, Sandia undermines its core advantage in the modern era. Ironic indeed.

In the environment that developed over the last quarter century, where money flowed into project work, Sandia had an advantage. Having employees chosen primarily for their ability to do homework worked out rather well. Project work is much like homework: well-defined, bounded, and carrying very little chance of failure. We are not doing research or finding breakthroughs. We are executing a project as defined, on schedule, against well-defined milestones and outcomes.

“Good ideas are not adopted automatically. They must be driven into practice with courageous impatience.” — Hyman G. Rickover

It is no wonder Sandia has grown so much larger in this period. Their approach is perfect for the modern era. The problem is that the modern era is one of a diminished United States. It is a country that has lost supremacy in scientific research and engineering and is ceding that crown to China. The USA has simply capitulated. China has simply pulled ahead through dull competence. A system that is better designed for the high control of people. In the meantime, the USA is sabotaging its prime advantage. Freedom of thought and bold solutions. It has given in to fear and distrust of institutions. Fear kills freedom. Trust is needed for bold action.

That said, I strongly suspect China is very much in the homework mode as well. It is a very controlling environment. The difference is that the Chinese have numbers on their side: vastly more researchers at vastly lower cost, free of much of the regulation we have here. They can efficiently execute more and bigger projects, and ultimately consume the advantage the United States once had. The combination of Chinese efficiency and American incompetence has blunted one of our major advantages. I just happened to be in an organization perfectly suited to thriving in a declining America.

“Bad news isn’t wine. It doesn’t improve with age.” — Colin Powell

Control is an attempt to keep issues and problems from boiling over. But problems do not magically solve themselves. A problem you put a lid on just gets worse and worse until it escapes containment. How many problems are we trying to control this way? The containment will fail. It always does. What will happen when they break free? Our current philosophy only ensures that the problems will be bigger and more damaging than necessary.

In the meantime, management acts out of fear and instills that fear in the people it manages. The fear revolves around control, around keeping everybody in line with the prime directive: keep the money flowing. It provides an element of stability to the labs, but it also saps them of inspiration and motivation, which are critical to executing the mission properly. In the process, we have no energy left to solve the major problems we face as a society. We are just allowed to nibble around the edges. The heart of the problems are never exposed.

Let me be perfectly clear about what this control sacrifices: the ability to confront and solve problems. Problems management doesn’t want to acknowledge won’t be solved, because they won’t be funded. The funding controls people, keeping them from doing what they think is right. This was the lesson of my last year at work. Instead, it focuses them on what they can be paid for. Management gets what it wants through power and force. Money is power today. It rules every decision. Right and wrong is measured in dollar signs.

“In any bureaucracy, the people devoted to the benefit of the bureaucracy itself always get in control, and those dedicated to the goals the bureaucracy is supposed to accomplish have less and less influence.” — Jerry Pournelle

In the process, problems that need solving simply languish. The real loss is the creative work necessary to solve them. That work leads to breakthroughs, and it also leads to many failures, because these problems are difficult, bordering on intractable. In a controlled environment, the people working on them don’t have the leeway to try bold ideas. For people to be bold, safety is needed. Control removes that safety. Without bold ideas, we are reduced to incrementalism at best and complete stagnation at worst. That is where we are today.

By failing to confront problems, we also fail to produce solutions. These solutions are the things that change the future, make it better. They are breakthroughs and discoveries. That is the other thing control denies us. Control is robbing us of progress and a more bountiful future. Fear does that. I will note that AI is a problem in need of vigorous and concerted effort. We need breakthroughs to help us navigate the perilous times ahead of us. Our control will deny us positive solutions.

“The prospect of domination of the nation’s scholars by Federal employment, project allocations, and the power of money is ever present — and is gravely to be regarded.” — Dwight D. Eisenhower

I recently met a friend at a conference who is quite critical of my writing. He sees it as self-destructive. Perhaps it is, in this sense: by writing about what has actually happened at the labs and what their problems are, I have burned my bridges. The labs aren’t interested in solving their problems. They are interested in hiding them. That is the essence of the control they seek. They are nervous about the future and about keeping their programs funded. They have concluded that the best way to manage that is to keep any problems under wraps.

I saw this in the messaging from leadership to the rank and file. Only success was talked about. Any issues or problems were only discussed in private. They could be mentioned in hushed tones without management acknowledgement. In public, everything was great. Success was the message. The organization is doing great, whatever the reality is. This is the nature of a toxic positivity that has taken root at the Labs. Maybe in the country as a whole. Problems lie festering in the shadows. They grow larger and more intractable in the fertile environment of enforced ignorance.

Control, and joining the project of power, brings money. I get several reactions to my writing. It is a service; it is self-destructive. It is patriotic; it is pointless. You give in so that you can have comfort. What I am describing is a low-key version of what is happening with Trump: power and money used to squash all resistance. That power and money is in service of decaying institutions and terrible outcomes. Trump’s version is over the top. Mine is quiet, but the mission is the same. People do shitty stuff to lead in a bad direction, and reward those who let them. It is self over community, self over the collective. It is dragging society down one capitulation at a time.

I burned those bridges by pointing out the problems, because pointing them out was the right thing to do. Being able to live with myself means doing the right thing instead of the easy thing. If I suffer for it, so be it. At least I have my integrity. Maybe at the end I can face myself and know that I did what the moment required.

“The day soldiers stop bringing you their problems is the day you have stopped leading them.” — Colin Powell

“For a successful technology, reality must take precedence over public relations, for nature cannot be fooled.” – Richard Feynman

I had the virtue of attending a scientific meeting when the news broke. There was a buzz when I arrived at the meeting. Since my morning routine was turned on its head, I had not digested the news before arriving. OpenAI claimed to have solved the Clay Prize Millennium Problem revolving around the existence of solutions to the Navier-Stokes equations. Parts of the entire story were the theater of the absurd. They used 10,000 agents working for nearly 90 hours to produce a 166-page proof that had been digested by another tool to confirm it. This is a big deal, and reactions from the scientific world were divergent.

I think it is important to step back first and put this whole thing into context. There are three major threads to consider:

1. OpenAI’s role and motivations are a basket of “red flags”.

2. The division of AI work and human labor is questionable.

3. The problem itself and the nature of the solution are questionable.

I will take these considerations one by one. Each of these is a different flavor of what the fuck?

“It’s a kind of scientific integrity, a principle of scientific thought that corresponds to a kind of utter honesty — a kind of leaning over backwards. For example, if you’re doing an experiment, you should report everything that you think might make it invalid — not only what you think is right about it.” – Richard Feynman

OpenAI’s Role and Character

Scientific discoveries do not happen in sync with the business or political cycle. This one did.

“What we are doing is finding ways for people to understand and think about mathematics.” – William Thurston

The fact that OpenAI was involved immediately projects a cloud of suspicion on the result. First and foremost is their leadership. Altman and Brockman are both extremely prone to lying and fabrication. They are leaders who should be met with vacant credibility. This is simply an empirical observation. They are trying to pump up OpenAI before their upcoming IPO and undercut Anthropic before theirs. This news does exactly that. Everyone should see that Altman is a sketchy as fuck character.

Secondly, the announcement came in close proximity to the release of their latest model, Astra. Coincidence? This seems unlikely. The whole thing feels very staged to boost this release. Red flag number two achieved!

Thirdly, the news cycle has been rightfully brutal to OpenAI. The Hugging Face hack by rogue agents was alarming. It was scary and damning to a company that had been devoted to AI safety. They appear to have lost that mission. There was another hack, and this news release successfully put a wet blanket on that report. Red flag number three is visible.

“If I have seen further it is by standing on the shoulders of Giants.” – Isaac Newton

Finally, the work is not intellectually separated from other efforts to solve this technical problem. There is not a clear separation from this other work, and OpenAI has been evasive. OpenAI tools were used to do some of this other work with genuine mathematicians. It is not clear that the information barriers are sharp. This would be some degree of plagiarism. The whole thing carries a stench of the shitty student cribbing an answer by peering over the shoulder of the smart student. Red flag number four is seen!

“The indiscriminate strip-mining of open problems for solutions may destroy the ecosystem from which the next generation of mathematical techniques, problems, and practitioners would have developed.” – Terence Tao

Each red flag needs to be addressed. Only the last two can be successfully refuted with some transparency. Doing so might begin to help rescue Sam Altman’s reputation. Fat chance!

“I am not accusing anyone of anything. I am stating what I was told.” – Tristan Buckmaster

The Division of AI and Human Labor

I am a relatively avid user of AI. I should disclose that I pay to use Claude. I also use other models without paying. I find ChatGPT and Grok are good with image creation (seen here). I also want to keep my context window with Claude clean and focused on more professional activities. I use Claude as an assistant for doing research. It is extraordinarily good at detail work, which I am sort of bad at.

For example, I used it to check my work on a recent scientific article. We had reviews, and two of the authors (including myself) used it to digest and check progress on the revised manuscript. One of the reviewers did a very sharp initial review. AI helped us work through every objection and checked our work. It is extremely and inhumanly pedantic. It assisted in getting to a complete and comprehensive revision. The paper was accepted this week. AI helped us, but we did the work.

“To create consists precisely in not making useless combinations and in making those which are useful and which are only a small minority. Invention is discernment, choice.” – Henri Poincaré

This gets to the issue of credit for the proof OpenAI claimed. In general, LLMs are giant nonlinear interpolation machines. They take vast stores of human knowledge in the form of language and digest it. They take prompts, then produce a reply that is highly probable. Is this new? The replies are unique because of the vastness of the models and data. This is akin to discoveries where well-defined knowledge is merged together to create something new. This is separate from the generation of brand new creative ideas. Creative and totally brand new ideas are uncommon. Merged ideas are the vast majority of breakthroughs.

The solution to the Clay Prize should be a work of creative discovery. The OpenAI work is almost certainly the nonlinear interpolation of existing human work. This lends circumstantial evidence to the worry of copying existing work. Furthermore, the ability to clearly define its sources and reasoning is damning. This is not a take that says AI should not be used for science or math. This is a take that says AI can and cannot do something. With the extreme complexity that this solution apparently has, disentangling this is itself a huge challenge.

“A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” – G. H. Hardy

This leaves the last consideration. Should we give a fuck about the result? Spoiler alert. Probably not.

“Standard methods from PDE appear inadequate to settle the problem.” – Charles Fefferman

The Question Itself

“There is a physical problem that is common to many fields, that is very old, and that has not been solved. It is not the problem of finding new fundamental particles, but something left over from a long time ago — over a hundred years. Nobody in physics has really been able to analyze it mathematically satisfactorily in spite of its importance to the sister sciences. It is the analysis of circulating or turbulent fluids.” – Richard Feynman

If you’re a regular reader, you know what I think. The Navier-Stokes Millennium problem is sort of stupid. It is not clear that it has anything to do with physical fluids. The rub is the incompressibility, which is unphysical. This causes the removal of sound waves, which now propagate at infinite (superluminal) speeds, violating causality. This is not to say that incompressible flows are not a useful approximation. It is. The question is whether it is useful for the nature of flows implied by the Clay Prize.

The prize revolves around finding out whether normal smooth solutions exist for all time. Why would we worry about this? Turbulence is the answer. A combination of theory due to Kolmogorov and observations indicates that turbulent fluids have singularities. This is the characteristic that the integrated rate of dissipation at large scales becomes independent of the value of viscosity. Viscous forces are the means of enforcing dissipation at small scales. It also keeps flows smooth at small scales.

The singularity that is implied is like a shock wave. You get a discontinuous jump in the fluid state. Shocks give the same character, where the large-scale structure and dissipation rate are independent of the exact value of viscosity. For turbulence, we are looking for something akin to this. The scaling in shocks and that posed by Kolmogorov are the same except for the leading constant, the jump in fluid state cubed. What I’ve questioned about incompressible flow is whether the connection to compressible (real) fluids retains this cubic behavior. It doesn’t.

“It is of some interest to note that in principle, turbulent dissipation as described could take place just as readily without the final assistance by viscosity.” – Lars Onsager

Now we get to the work here. The flows are driven by a source term that is apparently quite elaborate. It is likely that this source term is grossly unphysical. So we rightly ask whether this work with the source term actually solves the Clay Prize Problem at all. Full stop. Next we ask whether it has anything to do with turbulence. Unlikely. I don’t know the nature of the blow-up found. I hope it is a derivative singularity, as this would have some significance. If the blow-up is an infinite velocity, the result is truly not worthy of a single fuck. So the telltale sign of utility is the nature of the blow-up. If it is a derivative (or vorticity), it is worthwhile. If it is an infinite velocity, then the result is damning. The source term nature of the drive is discrediting a priori. At least with respect to physical fluids.

I look forward to learning more about all this in the coming weeks. The details matter immensely, but some of the aspects are engraved in stone.

“I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic.” – Horace Lamb

“Progress in science depends on new techniques, new discoveries and new ideas, probably in that order.” – Sydney Brenner

I’m going to revisit a paper that has a special place in my heart. It is a premature apex of a body of work terminated too early. A bit of the loss is my own fault as I moved to Sandia Labs. It was also fairly brutal to get published. It is a line of work that has potential, but the final development needs a boost. It is an investment in a stagnant area of research. There is also resistance from gatekeepers, who hold progress hostage. The important aspects of this are efficiency on realistic problems and selective preservation of extrema in the solution. The efficiency and verification aspects are progressing due to some recent effort.

I will elaborate on both below.

“You are more likely to be remembered by your expository work than by your original research.” – Gian-Carlo Rota,

Simply publishing the paper took some of the wind out of the sails of the results. This is all related to what determines the effectiveness of the method. As one might guess, this comes down to verification of results on discontinuous problems. As noted before, the accuracy of shocked problems is rarely computed in results. The results are purely judged qualitatively. Thus, fundamental beliefs and premises are not subjected to testing via quantitative verification. This is a community-wide accepted practice.

“The scientific paper in its orthodox form does embody a totally mistaken conception, even a travesty, of the nature of scientific thought.” – Peter Medawar

Accuracy is only considered quantitatively for smooth problems. In this way, one can demonstrate the formal order of accuracy. This formal order of accuracy does not determine the accuracy on problems with discontinuities. The entire field is focused on producing methods that can practically solve problems with discontinuities. These are virtually all practical applications we design the method to solve. In my view, this is an enormous community-wide gap. We only measure accuracy on problems that are immaterial to the applications.

A recipe for lack of progress!

It should come as no surprise that this is what we have got. Gatekeepers are removing progress, and “barbaric” new ideas are kept outside the walls of the high-resolution castle. The gatekeepers have ensured their legacies for now.

This gap in measurement is the central focus of this line of research. In my opinion, this is also the source of stagnation in methods’ development and progress.

To see the origin of my thinking, one needs to go back a few years. A seminal effort of my own was a collaboration with Jeff Greenough from Livermore. We studied the relative accuracy of a fifth-order weighted ENO method compared to a high-quality second-order Godunov algorithm. The knee-jerk response is that this is a futile contest. How can a second-order method compete with a fifth-order method? For smooth problems, it cannot. For problems with discontinuities, the answer is different, and the second-order method crushes WENO. This is especially true when cost is taken into account. WENO only wins in efficiency at enormous resolutions for problems replete with structure.

“Simplicity is prerequisite for reliability.” – Edsger Dijkstra

Details matter a lot in explaining this rather curious result. WENO methods typically use multistep time integration. This has two clear effects that determine most of the cost. The standard is a three-stage third-order Runge-Kutta integrator. Thus, the space discretization must be evaluated three times. The second impact is a factor of two decrease in time step size for similar CFL constraints. These two elements combine for a factor of six multiplier in cost for WENO in unit cost per mesh cell. For the most part, the spatial discretization cost is about the same per evaluation. As I will discuss later, any advantage for WENO can be eliminated by developing an adaptive method. The adaptive method in principle combines the lower-order method with WENO selectively. WENO is overly dissipative and inaccurate in monotone regions of the flow.

Another key aspect of this study is the nature of the low-order method. In the case of the piecewise linear method, the linear profile is optimally fourth-order accurate for the slope. In a similar fashion, the base PPM method uses a fourth-order edge value for each cell. In each case, monotonicity is checked and the approximation is modified if problems are found. Nonetheless, the high-order initial approximation endows the method with accuracy. This makes these methods far more competitive than a fully second-order approximation like Van Leer introduced. It is a lesson to apply going forward.

Add to this the effective accuracy of the two methods. On simple problems like Sod’s shock tube, WENO is half the accuracy of the second-order method. For the interacting blast waves, the accuracies are about the same. Finally, for the Shu-Osher problem at very late time, WENO is twice as accurate on a per-mesh-cell basis. The lesson being that WENO is only better if the problem has a great deal of structure and many extrema. To be more efficient requires extremely fine meshes. This motivates the development of a more adaptive discretization that accounts for this behavior.

“Since all models are wrong the scientist must be alert to what is importantly wrong. It is inappropriate to be concerned about mice when there are tigers abroad.” – George Box

The basic idea is to move between discretizations that are best for discontinuous solutions and those best for smooth extrema. One still needs to find and eliminate spurious oscillations. Separate these from physically resolvable oscillations. This is sort of a holy grail for numerical methods for hyperbolic PDEs. Some great work from the late 1990’s comes to the rescue. Two researchers from NASA Glenn (Suresh & Huynh) published a great paper on doing much of this work. First reproducing a monotonicity-preserving method and then added extrema detection and preservation to it. This was a great general-purpose approach that works for multistep methods like Runge-Kutta.

This work uses the median function developed and used by HT Huynh. In a sense, I fell in love with this function as a means of engineering methods. The method takes three arguments and returns the one bounded by the other two. Thus, it is great at enforcing bounds, which is the whole game for high-resolution methods. It also has the property that if two of the arguments are a certain order of accuracy, the output retains that order of accuracy. This is because the result is a convex combination of the other two, or one of those two. I have postulated that this result applies to linear and nonlinear stability properties too.

“Normal science, the activity in which most scientists inevitably spend almost all their time, is predicated on the assumption that the scientific community knows what the world is like… Normal science often suppresses fundamental novelties because they are necessarily subversive of its basic commitments.” – Thomas Kuhn

As a sidebar to the whole extrema preservation study, I rewrote the limiter for the PPM method using the median. Once the median is defined, the PPM limiter can be expressed in four median function calls. It is compact and IMHO far clearer than the classical PPM description, which is elaborate and somewhat non-intuitive. Looking at the (magical) median as a bounding function provides the necessary clarity of approach. Being parabolic in each cell, the PPM method could potentially support smooth extrema.

The simplest version of the basic idea is brutally simple. If the data is monotone and the approximation preserves this, simply use that. If it is not, then one can switch over to WENO, or some other extrema-preserving method. The cost of WENO is only taken where it matters. It also isolates WENO from the dissipative solution it gives for monotone circumstances. Under those conditions, WENO produces something akin to the dissipative minmod result. WENO is only superior with smooth extrema. This method improves on both methods. With PPM, I can make this a single-step method. It has a larger CFL and accelerates the WENO solution by a factor of six, plus sharpens the results in general.

“Perfection is achieved, not when there is nothing more to add, but when there is nothing left to take away.” – Antoine de Saint-Exupéry,

This unveils the basic approach for the advance. If some version of high-order works as a monotonicity-preserving approximation, use it. Do not do more work. If it does not, look at the data. Is it monotone or at an extremum? If it is monotone perhaps looks at other high-order, but lower than the initial attempt approximations. If one is at an extremum, determine if it is smooth and do something safe there. ENO or WENO are those sorts of approximations. If it is close to a shock, allow yourself to go to first-order. That’s it. It really is not that elaborate. It extends monotonicity methods to something more, and the verification work shows this. These methods are more accurate on a broad class of methods. In practice, this accuracy also makes them efficient. Colella and Sekora published a method with a lot in common with my ideas.

“My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful.” – Hermann Weyl

Now we get to the unfortunate part of the paper. The verification got removed in review. The associate editor demanded it rather bluntly: “If you want this published, take that shit out!” He and one of the reviewers, who is an asshole, killed this. I will note that the asshole reviewer is extremely good technically. Exceptional. Those are some of the most dangerous forms of assholes, by the way. I folded and removed the material. Now, 20 years later, I can finally get the basic idea of verifying shock tubes published. I can include a far better and more focused explanation and recipe for doing this. This includes a strong motivation in mathematics and efficiency from differences in accuracy.

I hope it helps alleviate the stagnation.

“Every genuine test of a theory is an attempt to falsify it, or to refute it.” – Karl Popper

The other idea that might have traction is the BVD and THINC-based methods. Combining this with high-order leads to impressive results. Imperfect, but hopeful. I was a reviewer of one of the articles, and felt like it was an arc of progress. I believe THINC might be the key here. It is a nonlinear first-order method that can deliver monotonicity with much less dissipation. The details are choosing the steepness parameter (and it having a parameter!). The other issue might be how the lack of first-order dissipation impacts the entropy condition. The question is whether the lack of dissipation in THINC ever threatens the entropy condition. It would be something good to study and put effort into.

“A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.” – Max Planck

References

Rider, William J., Jeffrey A. Greenough, and James R. Kamm. “Accurate monotonicity-and extrema-preserving methods through adaptive nonlinear hybridizations.” Journal of Computational Physics 225, no. 2 (2007): 1827-1848. (105 Citations)

Greenough, J. A., and W. J. Rider. “A quantitative comparison of numerical methods for the compressible Euler equations: fifth-order WENO and piecewise-linear Godunov.” Journal of Computational Physics 196, no. 1 (2004): 259-281. (85 Citations)

Suresh, Ambady, and Hung T. Huynh. “Accurate monotonicity-preserving schemes with Runge–Kutta time stepping.” Journal of Computational Physics 136, no. 1 (1997): 83-99.

Huynh, Hung T. “Accurate upwind methods for the Euler equations.” SIAM Journal on Numerical Analysis 32, no. 5 (1995): 1565-1619.

Colella, Phillip, and Michael D. Sekora. “A limiter for PPM that preserves accuracy at smooth extrema.” Journal of Computational Physics 227, no. 15 (2008): 7069-7076.

Deng, Xi, Yuya Shimizu, and Feng Xiao. “A fifth-order shock capturing scheme with two-stage boundary variation diminishing algorithm.” Journal of Computational Physics 386 (2019): 323-349.

Deng, Xi, Yuya Shimizu, Bin Xie, and Feng Xiao. “Constructing higher order discontinuity-capturing schemes with upwind-biased interpolations and boundary variation diminishing algorithm.” Computers & Fluids 200 (2020): 104433.

Rider, W. J., J. A. Greenough, and J. R. Kamm. “Combining high-order accuracy with non-oscillatory methods through monotonicity preservation.” International journal for numerical methods in fluids 47, no. 10-11 (2005): 1253-1259.

Rider, William J., and James R. Kamm. “How effective are high-order approximations in shock-capturing methods? Is there a law of diminishing returns?.” In Computational Fluid Dynamics 2004: Proceedings of the Third International Conference on Computational Fluid Dynamics, ICCFD3, Toronto, 12–16 July 2004, pp. 401-405. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006.

“Turbulence is the most important unsolved problem of classical physics.”– Richard Feynman

I’ve written several times about the lunacy of the incompressible Navier-Stokes Clay prize problem. Most of the issues relate back to incompressibility. I’ll summarize. Mathematically, incompressibility makes the equations an amalgam of all three types: elliptic, hyperbolic, and parabolic. With no viscosity, it is elliptic-hyperbolic. Physically, this implies infinite sound speeds. Thermodynamics is largely removed as well. It removes the most important source of nonlinearity from the equations — nominally, the mechanism for ubiquitous shock wave formation.

I will note that turbulence is utterly ubiquitous in science and technology. It is everywhere and happens quite naturally, being almost impossible to stop. The incompressible equations form singular solutions only with difficulty. Such solutions can be constructed, but it is hardly routine. This would be counter to the ubiquity of turbulence. Compressible equations, on the other hand, form them with ease. Here is the rub, and it is threefold:

1. Compressible flow is studied and focused on away from the zero Mach limit.
2. The behavior of compressible flow in that limit is poorly understood and poorly computed, especially for inviscid flows. Numerical solutions have deep pathologies.
3. The theory connecting incompressible and compressible flows is largely adiabatic and avoids key mechanisms.

“In mathematics you don’t understand things. You just get used to them.”– John von Neumann

Occam’s razor might take the ubiquity of turbulence to point to compressibility being intrinsic to turbulence. Singularities are readily found. The implications are big. The singularity is needed for the condition where the dissipation in the system is independent of viscosity. Only the large-scale (inertial range) flow determines the rate of dissipation. Shock waves and turbulent flows demonstrate this character. We know of the shock singularity, while the turbulent singularity is elusive. What if it was always there staring at us?

“Truth emerges more readily from error than from confusion.”– Francis Bacon

There have been studies of the mathematical link between compressible and incompressible flows. The key provision is the use of an adiabatic flow. The expansion is second-order in Mach number. The dissipation for turbulence and shocks is third-order. Thus, the flows are inviscid unless viscosity is explicitly included. This is a fatal flaw: the key inviscid mechanism leading to vanishing viscosity has been overlooked and eliminated. This is an unfortunate oversight. I think this is a general misunderstanding. The mechanisms for dissipation around singularities are third-order in Mach number (which is vanishing).

First is the wealth of existing solutions and theory, all revolving around incompressible flow. This would need to be discarded in large part. Compressible flows are less amenable to analytics. This is particularly true for non-adiabatic flows. My view is that the field is at an impasse and needs to break it with a big leap. Compressible and thermodynamic ideas might just be that.

Compressible flows have different mechanisms built in. These are compatible with observations. Singularity formation is the key, and the part of turbulence that cannot be ignored. Sound waves will always steepen and form singular structures, even at low Mach numbers, if the viscosity is small enough. It also injects the baroclinic term into the analysis, which might produce inhomogeneous vorticity. This would potentially change the mechanisms for some turbulence.

Shock waves are principally understood as a supersonic phenomenon, at Mach numbers above 1. Sonic booms and explosions are a couple of canonical examples people are given. The mechanism for waves to steepen into shocks does not go away. It is equally present at low speeds. Analysis has always been tied to incompressibility, with acoustics added on. In general, this has been applied to adiabatic situations. Turbulence is not this. It is fundamentally dissipative. To model turbulence, these equation sets need additional terms to be retained. The dissipative terms are generally one order higher.

Eddington said of a theory contradicting the second law that — “…there is nothing for it but to collapse in deepest humiliation.” – Arthur Eddington

The equations would need to retain the higher-order terms that cause dissipation due to nonlinear steepening. These are purely terms found with the acoustic modes, but they modulate the wave structure. Gradients are enhanced or diminished depending on the sign, and shock formation cannot be denied. Without dissipation, the flow will shock. It is also likely to be extremely complex and tied directly to the perturbations from unstable shear waves. Together, these should be an adequate and powerful engine for turbulence phenomena.

Perhaps the biggest change would come to the modeling of compressible turbulence. Today it is an add-on to incompressible theory: compressibility is an afterthought instead of the core notion. A fully compressible foundation would change this dynamic. It should lead to better results. It also makes more sense with the dynamics of implicit turbulence modeling. More importantly, it offers a new path for discovery and understanding. You have mathematics that supports a key mechanism for dissipation, the singularity formation.

“the unreasonable effectiveness of mathematics in the natural sciences”– Eugene Wigner

Researchers have nipped around the edges of this issue. The bridge that has failed is the direct connection. Shock dissipation is third order in velocity jump, leading to entropy creation. Turbulent dissipation is also third order in the velocity jump. This is called the longitudinal or normal velocity jump. This is the same scaling. Compressible shocks are one-dimensional. Kolmogorov’s results in the 4/5 law is three-dimensional. Connecting the two would marry this result to compressible flow and thermodynamics. Turbulence is perhaps the most pervasive and common means of entropy production. This would be quite satisfying for physics. Breakthroughs await.

Turbulence progress is stagnant. Something big needs to change for progress. This might just be the key to breaking the deadlock.