When Can World Models Recover Physical Laws?

Ye Yuan; Jun Liu

stat.OTSep 15, 2026

When Can World Models Recover Physical Laws?

Authors: Ye Yuan, Jun Liu

Organizations: School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan, China · Department of Applied Mathematics, University of Waterloo, Waterloo, Canada

Abstract

Accurate prediction does not establish that a world model has recovered a physical law: distinct dynamics can generate identical records under the same observation protocol. We formulate law recovery on a fixed physical domain under an explicit catalog of experiments, sensor uncertainty, and an acquisition budget. A rate--distortion converse separates the information needed to describe a law from the information the apparatus can reveal. Its constructive counterpart gives a finite response codebook and an explicit decoding budget. On compact world classes, uniform recovery is possible exactly when every pair of different laws is experimentally distinguishable; equivalently, the apparatus can recover all the entropy of every finite law source. An inverse response modulus quantifies stability. For Lipschitz fields on a $d$ -dimensional state--action domain, noisy full-state readouts after resets require minimax budget $Θ(\varepsilon^{-(d+4)/2})$ for squared law error $\varepsilon$ , compared with $Θ(\varepsilon^{-(d+2)/2})$ for direct field observations. Exact crossing-time symmetries establish the lower bound under adaptive experiment selection and arbitrary durations with constant inputs. Reproducible synthetic cases illustrate the separate roles of intervention, calibration, and repeated measurement. Together, the results identify which evidence supports a claim of physical-law recovery and the cost of acquiring it.

Figures & tables

Figure 1: Why accurate prediction of a recorded motion need not identify its physical law. Blue solid curves represent inverse-square attraction and orange dashed curves a linear restoring force. The laws are normalized to have the same inward acceleration

g_{0}

at radius

r_{0}

; position, acceleration, and time are measured in units of

r_{0}

g_{0}

, and

\sqrt{r_{0}/g_{0}}

. (a) The axes show the two planar position coordinates. Starting at radius

r_{0}

with tangential speed

\sqrt{r_{0}g_{0}}

gives the same circular trajectory under both laws, including the same timing. Their superposed curves are therefore observationally indistinguishable even with exact, continuous position measurements. (b) Acceleration magnitude versus radius reveals the hidden disagreement:

g/g_{0}=(r/r_{0})^{-2}

for inverse-square attraction and

g/g_{0}=r/r_{0}

for the restoring law. They meet at the observed radius but differ elsewhere. (c) A different preparation exposes that disagreement. Both trials now start at radius

2r_{0}

with zero velocity, and the vertical axis records radius as a function of time. The restoring law pulls the object inward faster because its initial acceleration is

2g_{0}

, compared with

g_{0}/4

for inverse-square attraction. These are noiseless trajectories. Section 7 adds Gaussian readout noise and quantifies the benefit of repeated releases.

Figure 2: Evidence that changes recoverability. Left: exact and simulated two-law selection error after release; repeating the circular orbit remains uninformative. Right: the exact uncalibrated Bayes floor and calibrated squared law error in the sensor-gain family. Shading is

\pm 1.96

standard errors over world averages; the dotted curve is the small-noise approximation. The panels report different risks, as labeled.

Release $R$	Readout $t$	Mean gap	Releases $n_{0.01}$	Cost $B_{0.01}$
1.0	0.6	0.017865	679	1765.4
1.5	0.4	0.082567	32	76.8
2.0	0.2	0.034863	179	393.8
2.0	0.4	0.137811	12	28.8
2.0	0.6	0.303986	3	7.8

Table 1: Cost of distinguishing the two orbit laws with error at most

1\%

. Each row repeats the same reset and one-readout episode, with

\sigma=0.1

. The cost is

B_{0.01}=(2+t)n_{0.01}

. These are analytically evaluated Gaussian tests, not fitted learning curves.

Figure 3: The same two physical laws can be easy, difficult, or impossible to distinguish under different readout designs. (a) The horizontal axis changes the release radius and the vertical axis changes the readout time. Color shows the separation-per-cost score ( 29 ) on a logarithmic scale; brighter regions give more separation for the charged cost. The star marks the largest score on the tested grid. The white curve is a set of equal terminal readings, although the intervening trajectories differ. (b) Exact Gaussian testing errors at the same total budget, including the integer number of affordable episodes. The gray dashed line is the uninformative circular preparation, and the horizontal dotted line marks $1\%$ error. Every curve uses the same known sensor noise $\sigma=0.1$ .

Figure 4: Recovering a continuous force parameter when the true world is not in the response codebook. (a) The release response

m(\theta)

varies with the force parameter. Orange circles are the 17 coarse codewords; teal crosses illustrate two off-codebook truths. All these laws agree on the original circular orbit. (b) Noise-averaged squared field error versus acquisition budget, for three codebook sizes at

\sigma=0.1

. Lines use Gaussian decision probabilities; markers and bars show Monte Carlo means and

\pm 1.96

standard errors, conditional on the same 32 true parameters. The bars may be smaller than the markers. Dotted lines show each decoder’s noiseless approximation error. (c) The same error for different sensor noise levels with 65 entries. Improving precision approaches the fixed codebook’s approximation level. These are finite-decoder studies in a known one-parameter family, rather than an empirical minimax result or a comparison of neural architectures.

Figure 5: Acquisition cost of the declared designs for state–action dimensions

d=1,2

. Costs include resets, readouts, and elapsed time. Shading is

\pm 1.96

standard errors over 64 noise replicates. Slopes use the five finest grids. The fixed-duration estimator retains differentiation bias. These curves illustrate estimator behavior on fixed laws; the minimax lower bound is proved separately.

World Model Learning

World model learning builds predictive models of how environments change under actions to support planning, simulation, and decision-making.

See all 419 papers

Stable and Counterfactually Robust Physical World Models from Imposed Structure and Learned Physics
Sep 24, 2026Yufeng Wang, Parivesh Priye, Lu Wei +1World Model Learning Physics-Informed ML
Stony Brook University · Georgia Institute of Technology · Westlake University
MultiEcho: An Experimental Science of Learned Worlds
Sep 27, 2026Meng Zhu, Airui ZhangCounterfactual Evaluation World Models
SHIHAO XINRUI Group
ALDER: Discovering the Laws of a World by Acting in It
Sep 27, 2026Teng Cao, Yu Deng, Quentin Delfosse +1Scientific Discovery Action-Conditioned World Models
Technische Universität Darmstadt · Intrinsic · CS Department, TU Darmstadt +2

Sep 24, 2026J/K move · Enter open · S save

Stable and Counterfactually Robust Physical World Models from Imposed Structure and Learned Physics

Yufeng Wang, Parivesh Priye, Lu Wei, Haibin Ling

Stony Brook University · Georgia Institute of Technology · Westlake University

World Model Learning Physics-Informed ML Causal World Models Hamiltonian Neural Networks Counterfactual Learning

Open paper AskPDF arXiv

A world model learns to forecast how a physical system evolves from recorded trajectories, yet the systems it imitates obey physical laws that are neither fully supplied nor reliably respected. The model may create energy, drift or diverge over long rollouts, and answer a changed law query using the law observed during training. We ask how much general physical structure must be hard coded into a world model, and how much system-specific physics can then be learned from data, for four properties to hold simultaneously: second law compatible dissipation, correct responses to interventions on physical parameters, stability out to one hundred times the training horizon, and robustness to disturbances. The imposed structure is general: dynamics are generated from the gradient of a learned energy through a fixed reversible operator, the energy is restricted to a confining class, a one way port can remove energy but never inject it, the drive channel is known, and the intervened parameter enters through a separable map. The model learns the energy functional, constitutive relations, dissipation rate, and couplings. Across an electromagnetic cavity, a particle in cell grid, and a shallow-water fluid, models with roughly nine thousand parameters recover constitutive functions with unit slope, separate conserving from dissipating worlds by four orders of magnitude using a single set of weights, and transfer changes in sign, magnitude, rate, and gravity to unseen values, where equal-capacity models without the same structure perform at chance or worse. A nonlinear constitutive law is recovered with its curvature preserved and predicts a held-out intervention

2

17\times

better than a converged linear model.

Abstract

World Model Learning

Explore similar work

Stable and Counterfactually Robust Physical World Models from Imposed Structure and Learned Physics

MultiEcho: An Experimental Science of Learned Worlds

ALDER: Discovering the Laws of a World by Acting in It