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

arXiv: https://arxiv.org/abs/2609.33728 · HTML: https://mlsanity.com/paper/2609.33728/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in World models for language agents, one of 11 directions in World models.
- Builds on 16 of its 22 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Placed in

- World models / [World models for language agents](https://mlsanity.com/wm/world-models-for-language-agents/)

## Builds on (most cited first, 16 of 16)

- [Dream to Control: Learning Behaviors by Latent Imagination](https://mlsanity.com/paper/1912.01603/) (2019)
- [D4RL: Datasets for Deep Data-Driven Reinforcement Learning](https://mlsanity.com/paper/2004.07219/) (2020)
- [Learning Latent Dynamics for Planning from Pixels](https://mlsanity.com/paper/1811.04551/) (2018)
- [Learning Complex Dexterous Manipulation with Deep Reinforcement Learning and Demonstrations](https://mlsanity.com/paper/1709.10087/) (2017)
- [robosuite: A Modular Simulation Framework and Benchmark for Robot Learning](https://mlsanity.com/paper/2009.12293/) (2020)
- [AI Feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularity](https://mlsanity.com/paper/2006.10782/) (2020)
- [LLM-SR: Scientific Equation Discovery via Programming with Large Language Models](https://mlsanity.com/paper/2404.18400/) (2024)
- [Model-Based Active Exploration](https://mlsanity.com/paper/1810.12162/) (2018)
- [RoboCasa365: A Large-Scale Simulation Framework for Training and Benchmarking Generalist Robots](https://mlsanity.com/paper/2603.04356/) (2026)
- [ODEFormer: Symbolic Regression of Dynamical Systems with Transformers](https://mlsanity.com/paper/2310.05573/) (2023)
- [SR-Scientist: Scientific Equation Discovery With Agentic AI](https://mlsanity.com/paper/2510.11661/) (2025)
- [Active Learning for Identification of Linear Dynamical Systems](https://mlsanity.com/paper/2002.00495/) (2020)
- [LLM-ACES: Closed-Loop Discovery of Dynamical Systems with LLM-Guided Adaptive Search](https://mlsanity.com/paper/2606.25039/) (2026)
- [Active Symbolic Discovery of Ordinary Differential Equations via Phase Portrait Sketching](https://mlsanity.com/paper/2409.01416/) (2024)
- [Kintsugi: Learning Policies by Repairing Executable Knowledge Bases](https://mlsanity.com/paper/2605.09487/) (2026)
- [Learning Explicit Behavioral Models with Adaptive Questions and World-Model Probes](https://mlsanity.com/paper/2606.07127/) (2026)

## Abstract (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.