The Physics Secret Behind AI "Solving" the N-Body Problem

Added: 2026-05-06

[00:00:00]If Newton's laws are so perfect, why can't we reliably predict the motion of just three objects for long? Honestly, these are equations simple enough to jot on a napkin. Gravity acts, objects accelerate, summarize it all, and you're finished. So, why does the universe start acting like a prankster the moment you add one extra planet?

[00:00:24]Because here is what you expect should happen. If I tell you exactly where everything is and exactly how fast everything is moving, you should be able to press fast forward on reality like it's a video.

[00:00:37]However, when we try to do this on a computer, something deeply unsettling happens. Tiny rounding errors, such as those smaller than a dust mote compared to a continent, may seem insignificant, but over time, they can compound. These small inaccuracies can cause simulated planets to behave abnormally.

[00:00:58]Orbits gradually weaken, bodies get slingshotted too forcefully, and some collide when they shouldn't. Others end up fleeing into a space. So, what are we missing? Let's start with the easy case, two bodies, a planet and a star. This is the clean classic physics problem.

[00:01:18]It's like a perfectly choreographed dance. One partner leads, the other responds, and the whole routine repeats.

[00:01:26]You can predict it ridiculously far into the future.

[00:01:30]Now, add a third body. Suddenly, it feels like a crowded wedding dance floor where three people simultaneously try to lead.

[00:01:39]Each a step influences what the others should do next, making the situation more chaotic.

[00:01:45]Worse still, the music, representing the pattern of pulls and tugs, keeps changing because it depends on everyone's current position. Here's the simple explanation. Each object attracts every other object. The strength and direction of these attractions depend on their current positions.

[00:02:04]Those positions change because of the pulls, which in turn alter the pulls.

[00:02:10]This leads to continuous shifts in positions and forces, and this is where a small errors become a big deal.

[00:02:17]Imagine at the step one, your simulation puts a planet just a hair to the left of where it should be. Step two uses that a slightly wrong position to compute gravity. So, now the acceleration is a slightly wrong. Then, a step three uses the slightly wrong velocity and position, so it's more wrong. And after thousands or millions of steps, you are off by a universe. This is the intuitive version of what people sometimes call a butterfly effect. Each moment is built on the last moment, and your mistakes breed. And computers, by their nature, make this harder because computers don't compute con- continuous motion. They can't. They take a snapshots. They do something more like stop-motion animation. Jump a tiny bit forward, recalculate forces, jump again, recalculate again.

[00:03:14]Even if those jumps are very small, you are still approximating a smooth curve with lots of little straightish hops.

[00:03:23]Over time, those hops may begin to behave in ways that seem almost supernatural. The simulation starts to generate energy or lose energy. Once your simulation begins leaking or creating energy, orbits begin to drift.

[00:03:40]The slingshot effects become exaggerated, and systems heat up or cool down without any physical cause.

[00:03:48]Eventually, this instability causes the entire simulation to break down.

[00:03:54]Before we go further, if you want to actually run the Hamiltonian neural net we are about to discuss us and see the actual math, I put together a free Jupyter notebook.

[00:04:05]It's linked in the description. Now, with that in mind, let's switch perspectives.

[00:04:11]How should we simulate motion for a long time without a slowly breaking the rules of physics?

[00:04:18]Maybe the problem is just the way we are writing physics down. Usually, we think of a simulation as just fitting into a spreadsheet. Each body, like the sun or the earth, is described by its position and velocity, and the list continues from there. But that view is kind of lonely. It makes each object feel like it's own little story. What if, instead, we looked at the system the way you would looked at a social network? Each object is a person, and each gravitational pull is a relationship line defining who influences whom. Now, the picture changes. Every planet is a dot, and between every pair of dots is a connection, an invisible a spring-like relationship whose a strength changes with distance. The closer two objects are, the more intense the interaction.

[00:05:14]The farther apart, the weaker the tug.

[00:05:18]And the whole system becomes this living web of influences.

[00:05:22]And in this web view, the important thing is how everything relates to everything else moment by moment. This is where a special kind of AI really shines, an AI designed to think about connections and relationships, often called a graph neural network, functioning as a kind of message-passing machine. Each object surveys its surroundings and sends out a small messages to others, such as, here is where I am, here is how fast I'm moving, and here is how a strongly I pull on you.

[00:05:57]Each object then receives messages from all others and update it next move based on its flood of information. This process reflects the structure of the real problem, in which many factors influence many other factors. But there is a problem. A typical pattern-finding AI is really good at guessing what comes next from examples, like predicting the next frame in a movie. And that can look amazing until you show it a situation that's even a slightly different from what it was trained on.

[00:06:31]Then, it starts hallucinating since it's not obeying the laws of physics.

[00:06:37]This right here is the application gap.

[00:06:39]You can train a GNN that looks good on a benchmark, but turning that into a stable model for real settings is a different skill. If you need to learn advanced machine learning systems for promotion or just for fun, you can apply to my personalized boot camp. Links in the description. All right. Now, let's bring in the missing rule the universe enforces. So, instead of a conventional AI, we need one that obeys the same unbreakable accounting rules the universe obeys.

[00:07:13]Here's the key cool. In a frictionless world, there is a quantity that acts like the universe's bank balance, energy.

[00:07:22]Energy can change its form. Some can be stored in position, like a ball held up high.

[00:07:29]Some can be stored in motion, like that ball flying through the air.

[00:07:34]But the total doesn't just magically appear or disappear. Think about it like a checking and savings.

[00:07:41]You can move money from checking to savings and back. You can invest it. You can break a $20 bill into smaller bills, but unless someone deposits or withdraws money, the total amount you own should stay the same. That's what energy does in idealized physics problems like planetary motion. It's the budget. An orbit is basically a constant negotiation between the speed and height. When a planet falls closer to the star, it speeds up, like it's converting height money into a speed money.

[00:08:18]When it moves farther away, it slows down, converting a speed back into height. But the budget stays balanced.

[00:08:26]Now, here is why simulation drift. If your calculator rounds things wrong every second, you don't notice at first, but after a million seconds, you're either mysteriously rich or mysteriously broke. Numerical drift is the same idea.

[00:08:44]What if we could develop an AI that is inherently resistant to bookkeeping errors?

[00:08:51]This concept draws on a fascinating principle from physics, in which a system can be represented by a single comprehensive ledger, a formula that calculates the total energy from positions and velocities.

[00:09:08]By adhering to this ledger's rules, the proper motion emerges automatically.

[00:09:14]Physicists usually refer to this fundamental energy formula as the Hamiltonian. Instead of asking the AI to directly predict the positions of objects, we can have it learn the energy ledger. From that, we can gently guide its understanding of motion, ensuring those ears stay nicely meshed.

[00:09:35]This way, instead of gathering more data, we are creating a smarter constraint that helps our AI model perform better.

[00:09:45]To prevent the butterfly effect, we not only analyze objects and how they interact with each other through a graph-based neural network, but we also pay close attention to all the signals passing through, making sure their updates align smoothly with the energy ledger.

[00:10:03]This comprehensive approach helps us maintain stability and consistency throughout the system.

[00:10:11]Picture a bank where each clerk works efficiently, yet before closing time, every desk must be perfectly balanced.

[00:10:19]Clerks aren't allowed to say, "It's close enough." or "I'll fix it tomorrow." If the accounts don't match, the update is rejected. And in physical terms, this act like guardrails.

[00:10:32]Normally, an AI simulator is like a car driving through fog, trying to stay on the road by copying what it saw in previous drives. It can do fine for a while, but drift a little, drift a little more, and suddenly it's off a cliff. With the energy constraint, the update rule becomes a track with barriers.

[00:10:57]The AI can still choose how to move inside the rails, but it can't jump the fence into the realm of impossible physics, where energy is created or destroyed out of nowhere. So, our AI is now explaining an energy landscape, which you can imagine as hills and valleys.

[00:11:18]Think of placing a ball on a hill.

[00:11:20]It doesn't require memorizing its future journey to understand what will happen.

[00:11:26]Instead, knowing the shape of the hill and trusting gravity to do its magic is enough.

[00:11:32]And here is why this helps with long-term chaos.

[00:11:36]Chaos means a small errors can explode, but when your updates are forced to conserve the budget, many of the worst runaway errors simply don't have permissions to grow.

[00:11:48]Your simulation can still be off, but its mistakes are constrained to motions that still the laws of physics.

[00:11:57]And this is another version of the application gap. It's one thing to follow the Hamiltonian argument intellectually, it's another thing to translate it into the kind of portfolio, interview stories, and project choices that actually get you hired. If you are targeting a data science role, I break that down step-by-step on my second YouTube channel.

[00:12:21]Link in the description. Now, let's test this idea where it matters. Speed and stability versus traditional solvers.

[00:12:30]So, we have introduced a new type of AI, but how does it compare to traditional methods?

[00:12:37]Classical physics solvers are like detailed accountants.

[00:12:41]They scrutinize every decimal place, recalculate frequently, and include additional substeps in each calculation to minimize errors.

[00:12:52]This makes them highly reliable.

[00:12:55]But this reliability comes at a cost, increased computation time. When the number of bodies grows from three to 3,000, the number of interactions increases exponentially. Now, let's compare that to the graph-based neural nets.

[00:13:12]Once it's trained, it quickly gets the hang of how these interaction webs change and grow.

[00:13:19]But does a graph-based neural net constrained by laws of physics remain stable over a million steps?

[00:13:27]A traditional method without a special care might slowly distort orbits.

[00:13:32]Circles become spirals. Systems gradually heat up or cool down for no reason. Eventually, something flies off or collapses.

[00:13:42]Meanwhile, the constrained model, the one forced to respect the ledger, keeps planets in believable paths much longer.

[00:13:51]Another advantage is generalization.

[00:13:54]It's similar to asking if AI can predict events it hasn't encountered before.

[00:14:00]A graph-based neural network understands relationships.

[00:14:04]So, adding more entities is like inviting more guests to the same type of event.

[00:14:11]Since the fundamental laws remain unchanged, the model can still operate effectively, even if it hasn't seen such a specific events before.

[00:14:21]Now, let's tie the threads together. The many-body problem becomes disastrous when tiny errors accumulate into fake energy, and fake energy destroys orbits.

[00:14:34]So, the resolution is surprisingly simple to say.

[00:14:38]By building the energy ledger into the model's decision-making, every step is forced to keep the books balanced. The simulation stops cheating over time.

[00:14:50]And there is a final metaphor I love for this.

[00:14:53]Ordinary AI trained to predict anybody motion is like drawing a map by copying what you saw out the window on past road trips.

[00:15:03]It might look convincing and even be accurate for familiar routes, but the moment you take a new turn, you are in trouble.

[00:15:12]Graph-based AI, which operates within the system's total energy limits, is much like learning the fundamentals of cartography. You understand how roads connect, how distances remain consistent, and how directions align.

[00:15:28]Even without having been to a place, your map still feels clear and logical.

[00:15:34]This approach helps us reduce a major source of chaos from our calculations, bringing more order and harmony to our processes.

[00:15:45]And the applications go well beyond just the movements of planets. Once you understand how to embed unbreakable rules into learning systems, you can tackle problems where maintaining long-term consistency is key, like molecules folding into proteins, fluids swirling into turbulence, or galaxies forming over billions of years. This also transforms how we understand trusting an AI model.

[00:16:14]Instead of thinking of it as a mysterious light box that simply makes impressive guesses, now it becomes more like a gray box, something partly learned, partly driven by rules. It's more akin to having a junior scientist who can show their work and explain their reasoning. From a scientific perspective, this is quite significant because it shifts the balance of the trade-off. Typically, you might face a choice between fast but unreliable systems or reliable but slow ones. However, rule-bound learning systems bring a friendly middle ground.

[00:16:56]They provide faster models that aren't just figments of imagination. This means you can explore countless what-if scenarios, such as varying initial conditions, parameters, or systems without endless delays and without the risk of the simulation quietly breaking down.

[00:17:17]So, although a rule-bound learning system is a still not an oracle that perfectly predicts the universe, it's something more practical. It's a new kind of simulation engine that stays faithful to the universe's accounting rules while scaling to systems too complex for brute-force calculation.

[00:17:40]Thanks for watching. Until the next video, take good care of yourself.