Driving · 2026-09-30
ALDER: Discovering the Laws of a World by Acting in It
Teng Cao, Yu Deng, Quentin Delfosse, Kristian Kersting
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TL;DR
ALDER discovers and revises explicit, equation-based world models by acting in the environment. It proposes parametric equation structures, fits coefficients, and uses an independent verifier on held-out data. When multiple hypotheses fit, an experiment selector queries cost- and safety-aware interventions where predictions disagree. Counterexamples update evidence and drive structural revisions; validated equations are used for prediction and control via an inverse problem.
Why it matters
The authors argue that accurate predictions on limited trajectories cannot identify the correct equation, and fixed candidate libraries cannot discover new laws. ALDER addresses both by combining (i) executable equation discovery and revision, (ii) independent admission on held-out data, and (iii) active experiment selection using disagreement between competing hypotheses, then using the validated model for goal-directed control.
Method
- Propose or revise parametric equation structures under a typed grammar; fit numerical coefficients with a numerical optimizer; evaluate candidates with an independent verifier on held-out data.
- Use an Active Rollout Selector that accounts for feasibility/cost/safety and selects interventions where candidate equations’ predictions disagree (prediction disagreement as an information-gain proxy).
- Iteratively add new experimental trajectories and failed predictions to an Evidence Ledger to revise equation structures; admit only models that pass protected validation; optionally use the validated equation for control via inverse action search.
Limitation
The authors state that the system refuses to return a model if no candidate passes validation when the process terminates, and that revision continues until a candidate passes, no distinguishing safe experiment remains, or the proposal or interaction budget is exhausted; the latter two cases return ⊥.
Abstract (from arXiv)
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.