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

Teng Cao; Yu Deng; Quentin Delfosse; Kristian Kersting

cs.LGSep 27, 2026

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

Authors: Teng Cao, Yu Deng, Quentin Delfosse, Kristian Kersting

Organizations: Technische Universität Darmstadt · Intrinsic · CS Department, TU Darmstadt · German Research Center for AI · The Hessian Center for AI

Abstract

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.

Figures & tables

Figure 1: Different world models can fit the same trajectories; Alder selects experiments to distinguish them. At mass $m_{0}$ , both equations fit the observations (A). At $2m_{0}$ with the same force, their predicted accelerations differ. The observation contradicts the mass-independent prediction (B).

Figure 2: Alder uses new observations to revise equations. The Law Revision Agent proposes an equation structure from public evidence, and the Coefficient Fitter estimates its continuous parameters. The Protected Admission Verifier checks fitted equations on held-out data; admitted equations can be used for prediction and action selection. After rejection, ARS may select a feasible experiment where candidate equations predict different outcomes; its observation guides the next revision.

A. Controlled discovery
	NovelLaw (60 cases)		PullCubeTool (5 runs)		PushT (5 runs)
Method	Functional recovery (%, $\uparrow$ )	Shift NMSE ( $\downarrow$ )	Admitted ( $\uparrow$ )	Shift NMSE ( $\downarrow$ )	Admitted ( $\uparrow$ )	Shift NMSE ( $\downarrow$ )
One-shot agent	43.3	$6.54\!\times\!10^{-2}$	0/5	$4.816\pm 1.973$	3/5	$0.842\pm 0.644$
Revision-only agent	75.0	$1.50\!\times\!10^{-2}$	0/5	$1.296\pm 0.024$	3/5	$1.292\pm 1.638$
Random-query agent	85.0	$9.15\!\times\!10^{-3}$	5/5	$0.194\pm 0.138$	4/5	$0.673\pm 0.473$
Alder	85.0	$\mathbf{5.58\!\times\!10^{-3}}$	5/5	$\mathbf{0.161\pm 0.052}$	4/5	$\mathbf{0.420\pm 0.185}$

Table 1: Interactive revision improves functional recovery; fitted equations predict ODEBase trajectories under shift. A: NovelLaw reports functional recovery (%) and median shifted NMSE over 60 cases; robot shifted NMSE is mean

\pm

SD over five episode-disjoint partitions. Rejected runs are included. B: median trajectory NMSE and mean symbolic accuracy on 59 ODEBase systems.

	OpenDrawer			OpenDoor
Model	OOD prediction RMSE ( $\downarrow$ )	Control MAE ( $\downarrow$ )	Success (%, $\uparrow$ )	OOD prediction RMSE ( $\downarrow$ )	Control MAE ( $\downarrow$ )	Success (%, $\uparrow$ )
Nominal	$6.8\pm 0.7$	$5.04\pm 0.54$	$83.6\pm 2.2$	$7.2\pm 0.8$	$4.26\pm 0.37$	$94.4\pm 2.0$
Passive law	$7.5\pm 1.0$	$5.78\pm 0.76$	$80.3\pm 3.6$	$7.4\pm 1.5$	$3.80\pm 0.49$	$89.7\pm 3.1$
Matched MLP	$3.7\pm 0.8$	$1.98\pm 0.35$	$96.9\pm 2.4$	$3.1\pm 0.7$	$1.81\pm 0.32$	$99.5\pm 1.1$
ALDER	$\mathbf{0.9\pm 0.1}$	$\mathbf{0.74\pm 0.10}$	$\mathbf{100.0\pm 0.0}$	$\mathbf{0.9\pm 0.1}$	$\mathbf{0.49\pm 0.06}$	$\mathbf{100.0\pm 0.0}$

Table 2: Explicit dynamics discovered by Alder improve shifted prediction while preserving downstream control performance in RoboCasa. Prediction RMSE and control MAE are in

10^{-3}

normalized joint-position units; success rates are percentages. Values are mean

\pm

SD across ten seeds, each averaging four mechanism families. Control uses 640 targets per model and task; bold denotes the best reported performance.

A. Missing structure or state
Regime	Admitted models	Localization AUROC ( $\uparrow$ )	Gated NMSE ( $\times 10^{-3}$ , $\downarrow$ )
Complete	100.0%	–	$0.042\pm 0.029$
Missing smooth	0.0%	$0.494\pm 0.019$	$14.5\pm 3.0$
Missing local	0.0%	$0.969\pm 0.007$	$18.5\pm 2.9$
Hidden local	0.0%	$0.997\pm 0.001$	$168\pm 114$

Table 3: Admission checks reject models with missing equation terms or state variables; degraded force observations reduce prediction coverage. A: AUROC and gated-residual NMSE are evaluated under distribution shift and do not determine admission. B: Drawer and Door are pooled; coverage is the fraction of worlds with admitted equations. RMSE uses those worlds only, whereas control success covers all targets, counting refusals as failures. Values with

\pm

are mean

\pm

SD.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Track	Variables	Additional operators	Output / limit
NovelLaw	distance, radial speed, masses, radii, radial action	protected division, literal power, exponential, $\tanh$	acceleration; 31 nodes
PullCubeTool	tool-frame cube pose, tool/cube velocities, relative axis, commanded pull	protected division, power, exponential, trigonometric functions, absolute value, maximum	pull-axis displacement; 95 nodes
PushT	T-frame pusher pose, no-object TCP displacement, swept-contact indicator	protected division, power, exponential, trigonometric functions, absolute value, maximum	angular displacement; 95 nodes
Door	current and initial hinge/latch angles	literal power, sine, cosine	displacement; 127 nodes
Peg	hole-frame $x,y,z$ and hole radius	literal power, absolute value, pairwise maximum	signed score; 63 nodes

Appendix

Table 4: Typed grammars expose building blocks, not complete target equations. In RQ1, the variables and primitive operators are available, but complete target expressions and harness-held reference bases are absent from the proposal sandbox.

	NovelLaw (60 cases)		PullCubeTool (5 runs)		PushT (5 runs)
Method	Functional recovery (%, $\uparrow$ )	Shift NMSE ( $\downarrow$ )	Admitted ( $\uparrow$ )	Shift NMSE ( $\downarrow$ )	Admitted ( $\uparrow$ )	Shift NMSE ( $\downarrow$ )
Baselines
Fixed/simple model	0.0	$5.94\!\times\!10^{-1}$	0/5	$5.773\pm 0.287$	0/5	$2.762\pm 1.098$
Polynomial SINDy	28.3	$1.16\!\times\!10^{-1}$	0/5	$16.573\pm 10.539$	0/5	$18.281\pm 18.708$
Matched MLP	0.0	$4.20\!\times\!10^{-1}$	0/5	$5.570\pm 0.407$	1/5	$1.815\pm 1.609$
Variants

Appendix

Table 5: Closed-loop revision improves functional recovery and shifted prediction over passive baselines. The full controlled comparison supplements Table 1 A. NovelLaw reports median NMSE over 60 cases; robot errors are mean

\pm

SD over five episode-disjoint partitions. Rejected runs are included. Bold marks the best result in each column, including ties.

Split	Structures	Cases	Structural role
Development	8	16	Covers every primitive component; used only for protocol and prompt development.
IID	4	12	Reuses a development motif with unseen coefficients and observations.
Compositional	12	36	Contains exactly one reserved component pairing.
Stress	4	12	Contains two simultaneous reserved component pairings.

Appendix

Table 6: NovelLaw tests new compositions of familiar primitives. Each test structure has three independently sampled parameter/data instances.

Method	IID	Comp.	Stress	All	Shift NMSE	Queries
One-shot	41.7	50.0	25.0	43.3	0.0654	0
Revision-only	58.3	86.1	58.3	75.0	0.0150	0
Random-query	91.7	91.7	58.3	85.0	0.00915	24
Alder	91.7	91.7	58.3	85.0	0.00558	24

Appendix

Table 7: Revised proposals recover more equations than one-shot proposals. RQ1 values are functional recovery (%); queries are medians over all 60 cases. Independent model calls make the last three rows an auxiliary sensitivity analysis, not a controlled selector ablation.

Comparator	Median shift ratio	Wilcoxon $p$	Recovery wins/losses
One-shot agent	0.0782	$5.18{\times}10^{-9}$	25/0
Revision-only	0.146	$2.66{\times}10^{-7}$	7/1
Random-query	0.901	0.203	3/3
Fixed library	0.00782	$1.63{\times}10^{-11}$	51/0
Polynomial SINDy	0.0249	$2.32{\times}10^{-10}$	34/0
MLP	0.0130	$1.80{\times}10^{-11}$	51/0

Appendix

Table 8: Paired tests support gains over one-shot and revision-only discovery. RQ1 shift-error ratios below one favor closed-loop Alder .

Model and evidence	PullCubeTool	PushT
Linear, public	5.773	2.762
Polynomial SINDy, public	16.573	18.281
MLP, public	5.570	1.815
Supplied structure, public	4.579	0.919
Supplied structure, all queries	0.212	0.588

Appendix

Table 9: Query evidence improves prediction even when structure is supplied. Hard-track entries are mean shifted NMSE over five folds; lower is better. “All queries” uses every initially unrevealed query outcome and is not an Alder discovery result.

	PullCubeTool			PushT
Method	Admitted	Shift NMSE	Shift MAE	Admitted	Shift NMSE	Shift MAE
One-shot	0/5	$4.816\pm 1.973$	$1.081\pm 0.368$	3/5	$0.842\pm 0.644$	$0.086\pm 0.032$
Revision-only	0/5	$1.296\pm 0.024$	$0.377\pm 0.032$	3/5	$1.292\pm 1.638$	$0.106\pm 0.072$
Random-query	5/5	$0.194\pm 0.138$	$0.144\pm 0.064$	4/5	$0.673\pm 0.473$	$0.078\pm 0.021$
Alder	5/5	$\mathbf{0.161\pm 0.052}$	$\mathbf{0.136\pm 0.029}$	4/5	$\mathbf{0.420\pm 0.185}$	$\mathbf{0.071\pm 0.024}$

Appendix

Table 10: Interactive revision improves model admission on both robot tracks. Values are mean

\pm

standard deviation over five folds. MAE is in centimeters for PullCubeTool and radians for PushT.

Acquisition	Identified ( $\uparrow$ )	Mean cost † ( $\downarrow$ )	Median cost † ( $\downarrow$ )
Passive sampling	4.4%	8.85	9
Random selection	94.0%	3.53	3
Geometric coverage	100.0%	1.91	1
Parameter uncertainty	99.5%	1.35	1
Alder (ARS)	100.0%	1.16	1
Oracle reference	100.0%	1.18	1

Appendix

Table 11: ARS identifies all targets with the lowest mean penalized cost. RQ2 uses 182 initially ambiguous cases, the same candidate models, and an eight-query budget. Bold marks the best non-oracle results, including ties.

Comparator	Mean cost reduction [95% CI]	Win/tie/loss
Passive	7.69 [7.55, 7.80]	182/0/0
Random	2.36 [2.05, 2.68]	151/29/2
Coverage	0.74 [0.56, 0.94]	63/118/1
Parameter uncertainty	0.18 [0.08, 0.32]	18/161/3

Appendix

Table 12: ARS reduces mean identification cost relative to each listed comparator. Paired RQ2 reductions are positive when they favor ARS.

Task	Selector	Identified	Support	Median queries
Drawer	Passive	50.0%	50.0%	3.5
	Random	100.0%	97.5%	0.5
	Coverage	100.0%	70.0%	0.5
	Uncertainty	100.0%	100.0%	0.5
	ALDER	100.0%	97.5%	0.5
Door	Passive	25.0%	25.0%	7.0

Appendix

Table 13: Structured queries identify Door models sooner than random queries. RoboCasa candidates are fixed. Queries are censored at seven after exhausting the six-query budget. “Support” reports exact recovery of the optional equation terms after coefficient thresholding.

	OpenDrawer		OpenDoor
Model	OOD prediction RMSE ( $\downarrow$ )	Relative success (%, $\uparrow$ )	OOD prediction RMSE ( $\downarrow$ )	Relative success (%, $\uparrow$ )
Nominal	$0.0068\pm 0.0007$	$96.7\pm 1.4$	$0.0072\pm 0.0008$	$95.9\pm 2.0$
Passive law	$0.0075\pm 0.0010$	$94.2\pm 2.1$	$0.0074\pm 0.0015$	$96.4\pm 2.4$
Matched MLP	$0.0037\pm 0.0008$	$99.5\pm 1.1$	$0.0031\pm 0.0007$	$\mathbf{100.0\pm 0.0}$
ALDER law	$\mathbf{0.0009\pm 0.0001}$	$\mathbf{100.0\pm 0.0}$	$\mathbf{0.0009\pm 0.0001}$	$\mathbf{100.0\pm 0.0}$
Parameter oracle	$0.0009\pm 0.0001$	$100.0\pm 0.0$	$0.0009\pm 0.0001$	$100.0\pm 0.0$

Appendix

Table 14: Equation-based models improve prediction under the original finite-action evaluation. Values are mean

\pm

SD over ten seeds, each averaging four dynamics families. Relative success means executed target error within 0.01 of the best finite candidate’s error; it is not the absolute target-reaching success in Table 2 . Each task has 640 targets per model. Bold marks the best non-oracle results, including ties.

Regime	Equation only	MLP	Always residual	Gated residual
Complete	$4.20{\times}10^{-5}$	0.0109	0.000345	$4.20{\times}10^{-5}$
Missing smooth	0.217	0.0133	0.00380	0.0145
Missing local	0.0485	0.0308	0.0191	0.0185
Hidden local	0.108	0.134	0.183	0.168

Appendix

Table 15: Residual correction helps missing-term regimes but not hidden-state failures. RQ4 reports shifted prediction NMSE; lowest error is bold.

Passive Symbolic Discovery
Method	Recon. NMSE $\downarrow$	Gen. NMSE $\downarrow$	OOD NMSE $\downarrow$	Complexity	Sym. Acc (%) $\uparrow$
SINDy	5.07e-04	5.09e-01	1.41e+00	11.7	18.5
Operon	7.95e-05	2.57e-01	1.84e+00	14.0	2.4
PySR	2.84e-03	8.82e-01	1.43e+00	6.6	18.1
E2E	3.69e-01	1.49e+00	2.19e+00	52.6	0.0
ODEFormer	4.61e-03	3.83e-01	2.04e+00	15.9	16.5

Appendix

Table 16: Alder achieves low median error across ODEBench prediction settings. We report the median NMSE (lower is better) across reconstruction, generalization, and future-time OOD prediction, mean symbolic accuracy (higher is better), and mean expression complexity (ODEBench mean = 19.3). Best NMSE and symbolic-accuracy results are in bold, and second-best results are underlined. The Alder row is shaded.

Method	Public dev.	Protected ID	High-angle OOD	Admitted
Zero displacement	6.671	6.501	28.866	0%
Linear joints	0.810	0.813	11.660	0%
Polynomial SINDy-3	0.025	0.027	1.455	40%
MLP	0.047	0.051	10.546	0%
Additive trigonometric library	0.068	0.065	2.626	0%
Full composed reference	0.020	0.021	0.568	100%

Appendix

Table 18: One-shot joint equations extrapolate to unseen Door angles. Adroit Door uses five episode partitions. Errors are handle-

x

MAE in cm; lower is better. OOD contains only unseen high-angle states.

Method	Public dev.	Protected ID	Radius OOD	OOD success F1	Admitted
Always failure	0.500	0.500	0.500	0.000	0%
Linear	0.557	0.553	0.552	0.266	0%
Polynomial logistic-3	0.769	0.767	0.754	0.450	0%
MLP	0.905	0.901	0.796	0.605	0%
Additive absolute rule	0.649	0.646	0.638	0.369	0%
Full non-smooth reference	1.000	1.000	1.000	1.000	100%

Appendix

Table 19: One-shot constraints generalize across held-out peg-hole radii. PegInsertionSide provides binary simulator labels. All metrics except “Admitted” are balanced accuracy or success F1; higher is better.

Backbone	Total/active B	Admitted	Support	Queries	Pred. cover	NMSE	Tokens
GPT-5.6 (primary)	not public	100.0%	83.3%	0.5	100.0%	$7.85\times 10^{-8}$	345788
Qwen3.5-9B	9/9	100.0%	66.7%	1.0	100.0%	$3.34\times 10^{-7}$	35082
GPT-OSS-20B	21/3.6	100.0%	41.7%	1.0	100.0%	$3.16\times 10^{-6}$	43037
Qwen3.8-27B	27/27	91.7%	75.0%	0.5	100.0%	$8.77\times 10^{-7}$	56292
Qwen3.6-35B-A3B	35/3	91.7%	41.7%	1.0	100.0%	$3.93\times 10^{-6}$	54832
Qwen3-Coder-Next	80/3	58.3%	33.3%	1.5	100.0%	$5.85\times 10^{-5}$	78094

Appendix

Table 20: Alder discovers validated equations with multiple open-weight backbones. On 12 matched RoboCasa worlds, the first row is the primary backend; the remaining seven are open-weight ablations. Queries are medians with non-admitted runs assigned budget plus one; NMSE is the geometric mean over finite predictions. Total/active parameter counts follow public model declarations, while the primary parameter count is not public. Tokens are API-reported mean request totals and are not a quality-normalized cost metric. “Support” uses the same strict optional-term recovery criterion as the primary study.

Method	Calls	Proposals	Queries	Input tokens	Agent h
Revision-only	60	138	0	31.8M	1.97
Random-query	60	117	1,368	30.9M	2.15
Alder	60	125	1,560	34.1M	2.22

Appendix

Table 21: Each full agent variant uses one invocation per case. RQ1 usage is summed over 60 cases. One-shot is derived from Alder ’s stored first round and incurs no additional call.

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