When Can World Models Recover Physical Laws?

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.

World models can be studied as experimental systems with response laws of their own. We introduce MultiEcho, a framework for estimating these laws through controlled counterfactual interventions, delimiting their applicability, and separately testing their physical correspondence. Across nine simulated physical systems and seven frozen model configurations, three-reference estimators predict complete intervention responses and recover intervention parameters. Estimator selection uses discovery data only; frozen fits are evaluated on validation and confirmation contexts. The experiments distinguish response predictability, intervention readability and physical accuracy. Responses can be locally describable yet poorly match physical effects in the same target coordinates. Event-window, visibility and camera interventions reveal conditional applicability, and paired generator configurations show reduced readability under a scene prompt with stronger guidance. Magnitude sweeps expose small image errors alongside large relative effect errors. An exact-reset material experiment separates registered visible-response success from fixed-readout failure on material-dependent futures at matched positions and velocities. Exact finite-scale identities resolve odd and even response errors; first-order remainder bounds specify when refined calibration converges. MultiEcho provides an experimental basis for studying learned-world laws independently of, and in relation to, physical laws.

Reliable world models should not only predict future states but express how actions change the world in an explicit, transparent and testable form, such as equations. Yet methods that rely on a fixed set of trajectories cannot distinguish equally good competing hypotheses, while searches over a fixed set of predefined candidates cannot discover equations outside the initial hypothesis space. We introduce ALDER (Action-guided Law Discovery, Evaluation, and Revision), a method that actively proposes novel experiments to test and revise models. Specifically, ALDER proposes parametric equations; a numerical optimizer fits their coefficients; an independent verifier tests these candidates on held-out data. To distinguish between competing valid hypotheses, a cost- and safety-aware selector queries interventions, in the form of novel experiments. The resulting counterexamples update the evidence ledger and guide the next structural revision, while incompatible laws are discarded. Across an in-house benchmark, ODE equation discovery tasks, and robotic experiments, ALDER discovers laws beyond its initial formula set, repairs failed model proposals, distinguishes fixed candidate models with fewer interactions, and improves out-of-distribution prediction. Furthermore, given a current state and a target, ALDER selects control actions by solving the inverse problem defined by its validated world model. Together, these results show that explicit equation-based world models can be tested and revised through interaction, then naturally used to guide goal-directed control.