🤖 machine learning

Prediction Is Not Physics: Learning and Evaluating Conserved Quantities in Neural Simulators

This paper demonstrates that while standard neural simulators often fail to preserve physical conservation laws despite accurate trajectory predictions, specialized architectures like Conservation Discovery Networks can successfully learn globally conserved quantities from Hamiltonian trajectories, though their performance critically depends on training strategies such as temporal consistency and alignment with initial conditions.

Original authors: Andrew Bukowski, Aditya Kothari, Simba Shi, Ishir Rao

Published 2026-05-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Andrew Bukowski, Aditya Kothari, Simba Shi, Ishir Rao

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are teaching a robot to play a game of billiards. You show the robot thousands of videos of balls bouncing off each other. The robot gets really good at predicting where the balls will be one second later. It's so good that if you ask it, "Where is the red ball in 0.1 seconds?" it answers with near-perfect accuracy.

But here is the catch: The robot doesn't actually understand the rules of physics.

This paper, titled "Prediction Is Not Physics," explores a strange problem: A computer model can be excellent at predicting where things go, but terrible at understanding why they go there. Specifically, it fails to respect the law of Conservation of Energy.

The Core Problem: The "Drifting" Robot

In the real world, energy doesn't just appear or disappear. If you throw a ball, its speed and height trade off perfectly. If it goes higher, it slows down. The total "energy budget" stays exactly the same.

The researchers trained a "Diffusion Model" (a type of AI that generates motion) on these physics simulations.

The Good News: The AI predicted the ball's path with very low error. It looked right.
The Bad News: When they checked the energy, the AI's "energy budget" was going crazy. In some cases, the AI's energy fluctuated 36,000 times more than it should have.

The Analogy: Imagine a bank account. The AI is a cashier who can perfectly predict how much money you will have in your pocket next week. But, the AI's internal ledger shows your balance jumping from $10 to $10,000 and back to $0 randomly, even though you never deposited or withdrew anything. The prediction of where you are is right, but the physics of your wealth is broken.

The Experiment: Can the AI Learn the Rules?

The researchers asked: Can we teach an AI to discover the "Conservation Law" (the rule that energy stays constant) just by watching the motion, without telling it the math beforehand?

They tested three different types of "detectives" (AI models) on three classic physics problems:

A thrown ball (Projectile motion).
A swinging pendulum.
A bouncing spring.

Detective 1: The "Structured" Architect

This AI was built with a specific blueprint: "You must calculate energy as Speed Energy + Position Energy."

Result: It was perfect. It found the energy rule almost instantly.
Weakness: It was rigid. When the researchers added "noise" (like static on a TV screen) to the data, this detective got confused. It tried to fit the noise into its strict blueprint and made mistakes.

Detective 2: The "Black Box" (CDN)

This AI had no blueprint. It was just a neural network told: "Find a number that stays the same as the object moves, but don't just guess '5' for everything."

Result: It was very good, but it needed a tiny hint. If the researchers gave it a weak nudge at the start (telling it, "Hey, your first guess should look a bit like real energy"), it performed almost as well as the Structured Architect.
The Surprise: When the data was noisy, this flexible detective actually did better than the rigid one. It could ignore the static and find the true pattern.

Detective 3: The "Polynomial" Student

This AI tried to find the rule by mixing simple math formulas (like $x^2$ , $\sin(x)$ , etc.).

Result: It was very sensitive to how long it studied.
- Short Study: It gave up early. It found a "partial" rule (like only looking at speed) and got a low score (0.78).
- Long Study: With more time and data, it figured out the full, complex rule and got a near-perfect score (0.9998).
Lesson: Sometimes, a model isn't "bad"; it just hasn't been trained long enough to escape a "local trap" where it thinks it has found the answer.

The Big Takeaway

The paper concludes with a warning for anyone building AI for physics:

"Being accurate at predicting the next step is not the same as understanding the laws of the universe."

You can have an AI that predicts the future perfectly well (low error) but violates the fundamental laws of nature (energy drifts wildly). The researchers found that simply telling the AI to "be conservative" isn't enough; you often need to guide it with a hint of the real physics, or ensure it trains long enough to escape bad solutions.

Summary in a Nutshell

Prediction $\neq$ Physics: An AI can guess where a ball goes without knowing energy is conserved.
The "Drift" is Huge: Even accurate AI can have energy errors thousands of times larger than reality.
Flexibility vs. Rigidity: Rigid models (built with physics rules) are great on clean data but break with noise. Flexible models (learning from scratch) are tougher against noise but need more training time to find the right answer.
The Lesson: To simulate the real world, we need to measure not just "how close is the prediction?" but "does the energy stay constant?"

Technical Summary: Prediction Is Not Physics: Learning and Evaluating Conserved Quantities in Neural Simulators

Problem Statement

Neural networks trained on physics simulation data often achieve high short-term prediction accuracy (low rollout Mean Squared Error) while failing to obey fundamental physical laws, such as the conservation of energy. Even when generative models like diffusion networks are guided by learned conservation structures during sampling, they frequently produce trajectories where energy drifts significantly over time. This disconnect raises a critical question: Can neural networks learn or select globally conserved quantities directly from state observations, and is low prediction error sufficient to guarantee physically valid motion?

The authors investigate whether models can identify invariant properties from physical trajectories to verify models, guide generation, or identify physical laws, specifically addressing the gap between trajectory-level accuracy and global conservation.

Methodology

Environments and Data

The study benchmarks three classical Hamiltonian systems with known analytical invariants:

Projectile Motion: Separable energy $E = \frac{1}{2}(v_x^2 + v_y^2) + gy$ .
Nonlinear Pendulum: Non-separable Hamiltonian $H = \frac{1}{2}\omega^2 - g \cos \theta$ .
Spring-Mass System: Separable energy $E = \frac{1}{2}kx^2 + \frac{1}{2}v^2$ .

Data is generated using exact closed-form solutions or adaptive RK45 integration to ensure ground-truth energy drift is negligible ( $10^{-5}$ to $10^{-7}$ ). Experiments include noise-free trajectories and trajectories with 1% additive Gaussian noise.

Models Evaluated

The authors compare four approaches to learning invariants:

Structured Energy Network (SE): An architecture explicitly decomposing the scalar into kinetic and potential terms, $H(s) = T_\theta(v) + V_\phi(q)$ . This serves as a baseline with strong physical priors.
Black-box Conservation Discovery Network (CDN): A standard MLP with no imposed physical structure.
- CDN-Conservation: Trained only on temporal consistency.
- CDN-Aligned: Trained on temporal consistency plus a weak standardized alignment loss to analytical energy at $t=0$ .
Polynomial CDN: A linear model $f(s) = w^\top \phi(s)$ where $\phi(s)$ includes degree-2 polynomials and trigonometric features.
Diffusion Baseline: A DDPM trained to generate state changes ( $\Delta s$ ), guided during rollout by the CDN invariant and a soft projection toward the SE network's energy level set.

Loss Functions

The primary objective for invariant learning combines two terms:

Temporal Consistency: Minimizes the difference between the learned invariant at consecutive time steps, $f(s_{t+1}) - f(s_t)$ .
Variance Hinge: Prevents the trivial solution where the model outputs a constant for all inputs by enforcing variance across the initial batch.
Alignment Loss (Optional): A weak standardized loss at $t=0$ to align the learned invariant with the analytical energy up to scale and shift.

Key Contributions and Results

1. Prediction Accuracy Does Not Imply Conservation

The diffusion model achieved low rollout MSE (e.g., $3.4 \times 10^{-4}$ for the pendulum) but failed to conserve energy. The standard deviation of energy in generated trajectories was 7,500 to 36,000 times larger than in ground-truth trajectories. This demonstrates that low prediction error and soft invariant guidance are insufficient to enforce global conservation laws.

2. The Necessity of Weak Supervision for Invariant Selection

While the conservation objective (temporal consistency) encourages the model to learn something constant, it is underdetermined.

CDN-Conservation (No Alignment): Without alignment to analytical energy, the model fails to recover the true Hamiltonian. On the pendulum and spring-mass systems, the Pearson $R^2$ collapsed to near zero ( $< 10^{-3}$ ).
CDN-Aligned: With a small alignment term ( $\lambda_{align}=0.2$ ), the black-box CDN achieved $R^2 \ge 0.996$ across all systems, proving that temporal consistency combined with weak supervision can recover physically meaningful quantities without structural priors.

3. Sensitivity to Training Schedules (Polynomial CDN)

The polynomial CDN exhibited significant sensitivity to training duration and data volume:

Short Schedule: On the pendulum, it converged to a partial solution dominated by the kinetic energy term ( $\omega^2$ ), missing the potential term ( $\cos \theta$ ), resulting in $R^2 = 0.78$ .
Long Schedule: With more epochs and data, the model recovered the full Hamiltonian, reaching $R^2 = 0.9998$ .
This indicates that the conservation loss landscape contains poor local minima (partial Hamiltonians) that require sufficient training budget to escape.

4. Robustness to Noise

Under 1% additive Gaussian noise, the Structured Energy Network's performance degraded uniformly ( $R^2$ dropped by $\approx 0.013$ ). In contrast, the Black-box CDN was more robust on the projectile and spring-mass systems, outperforming the structured model. The authors suggest that forcing a $T(v) + V(q)$ decomposition on noisy data requires fitting corrupted signals separately, whereas the unconstrained CDN can learn a robust function of the full state.

5. Symbolic Regression Validation

Using STLSQ sparse regression on the analytical energy targets, the authors confirmed that their feature library was expressive enough to recover exact symbolic equations for all three systems. However, they note this is a pipeline check rather than unsupervised discovery, as the regression target was the known energy.

Significance and Claims

The paper argues that trajectory-level accuracy metrics (like MSE) are insufficient benchmarks for physical simulation models. A model can be highly accurate in the short term while violating fundamental conservation laws.

The authors claim that:

Conservation is a distinct axis of quality: It must be evaluated separately from prediction error.
Architectural priors have trade-offs: While structured networks (SE) perform near-perfectly on clean data matching their assumptions, they may be more brittle to noise than flexible black-box models (CDN) when weak supervision is provided.
Optimization matters: The ability to recover conserved quantities depends heavily on training schedules; limited budgets can lead to partial solutions that appear valid (moderate $R^2$ ) but fail to capture the full physics.
Guidance is not a cure-all: Soft guidance from learned invariants during diffusion sampling does not guarantee energy conservation over long rollouts, highlighting the need for stricter constraints or better sampling strategies.

The work concludes that while neural networks can learn invariants, doing so reliably requires careful consideration of loss design, training duration, and the balance between structural priors and model flexibility.