See, Hypothesize, Validate: Multimodal Agentic Framework for Discovering Governing PDEs

Sarang Manoj Pekhale; Amartya Roy; Rajat Sarkar; Souvik Chakraborty

cs.AIAug 28, 2026

See, Hypothesize, Validate: Multimodal Agentic Framework for Discovering Governing PDEs

Authors: Sarang Manoj Pekhale, Amartya Roy, Rajat Sarkar, Souvik Chakraborty

Organizations: Department of Applied Mechanics Indian Institute of Technology Delhi, India Yardi School of Artificial Intelligence (ScAI) Indian Institute of Technology Delhi, India

Abstract

Discovering governing partial differential equations (PDEs) from observational data remains a core challenge across the sciences. Existing sparse-regression, symbolic-regression, and LLM-based approaches can be constrained by predefined libraries, noise sensitivity, hallucination, or limited iterative refinement. We introduce MAGE (Multimodal Agentic Governing Equation Discovery), an agentic framework that organizes PDE discovery as a confidence governed hypothesis validation loop inspired by the scientific cycle of observation, hypothesis, and falsification. Four role-specialized agents collaborate: a Differential Observer computing derivatives and diagnostic visualizations; a VLM-powered Phenomenology Extractor distilling qualitative cues from multimodal diagnostics; an LLM-driven Governing Law Synthesizer proposing candidates without a predefined library; and an Equation Arbiter fitting coefficients and assigning confidence scores. Discovery iterates until the top candidate clears a user-specified threshold, providing a structured process with an explicit accept-reject protocol. On the evaluated canonical PDE suite, MAGE obtains 8/8 exact structural recovery and the lowest coefficient error among the compared methods on 7/8 systems, with improvements of up to 4 orders of magnitude and a geometric-mean improvement of approximately 3 orders of magnitude. The pipeline also recovers the expected operators in two complex geometries and, on one laboratory sensor record, selects a cubic restoring-force model with held-out $R^2=0.98538$ . These results support further study of structured agentic reasoning for library-free governing-law discovery, while broader generalization remains to be evaluated.

Figures & tables

Figure 1: MAGE Framework. Four specialized agents communicate via a Shared Chat Memory . Agent 1 computes gradients and spectral features from raw data. Agent 2 uses a VLM to identify dominant physical regimes and outputs a JSON evidence report. Agent 3 conditions an LLM on this evidence to propose $N\!=\!10$ candidate PDEs $\{E_{0},\ldots,E_{9}\}$ . Agent 4 evaluates candidates symbolically; if confidence $\geq\!80\%$ the top equation is returned, otherwise the loop restarts from Agent 1 for up to three iterations.

Figure 2: End to end flow of the multimodal agentic framework MAGE on the NLS benchmark. The user/system controller routes artifacts between specialized agents: diagnostic visual evidence from Agent 1, a JSON evidence file from Agent 2, candidate equations from Agent 3, and executable validation from Agent 4. Detailed prompts and intermediate outputs are provided; refer to Appendix.

Aggregate summary		PF	SGA	WS	WI	PSR	PR	L4E	EG	MAGE
Exact (%)		42.9	14.3	28.6	42.9	0.0	0.0	85.7	71.4	100.0
Fail (%)		57.1	85.7	71.4	57.1	100.0	100.0	14.3	28.6	0.0
$\overline{\varepsilon}$		5.04e-2	4.70e-2	2.18e-2	8.90e-4	2.49e2	4.49e4	3.13e-2	6.93e-2	7.96e-7
Characteristic		Spur.	High- $\varepsilon$	Unstable	Sensitive	Severe	Failed	Strong	Moderate	Best tested
Per-equation results		Method
Equation	Metric	PF	SGA	WS	WI	PSR	PR	L4E	EG	MAGE

Table 1: Aggregate and per-equation quantitative comparison on the canonical PDE benchmarks.

\overline{\varepsilon}

is the geometric mean of the eight displayed per-equation normalized RMS coefficient errors

\varepsilon

, including errors for structurally incorrect recoveries; exact zeros are replaced by

10^{-15}

for logarithmic averaging. MT and ET denote missing and extra term counts. Lower is better for all three error metrics, and bold indicates the best value in each row. PF: PDE-FIND; WS: W-SINDy; WI: Weak-Ident; PSR: PINN-SR; PR: PDE-READ; L4E: LLM4ED; EG: EqGPT.

Panel of Figure 3 — Figure 3: PDE discovery in complex geometries.

	Coefficient error $\bm{\varepsilon}$						Structural error ( MT $+$ ET )
Equation	0%	1%	2%	5%	10%	15%	0%	1%	2%	5%	10%	15%
NLS	8.10e-5	1.2e-4	2.4e-4	1.21e-3	3.63e-3	6.34e-1	0	0	0	0	0	0
PDE-Divide	2.36e-4	1.71e-2	2.96e0	3.10e0	3.09e0	3.34e0	0	0	3	3	5	5
KdV	4.93e-6	1.0e-4	7.5e-4	7.95e-3	3.28e-2	1.44e-1	0	0	0	0	0	0
KS	1.73e-6	9.1e-4	1.75e-3	5.12e-3	1.79e-3	2.16e-1	0	0	0	0	1	0
Burgers	8.0e-6	1.46e-3	3.31e-3	1.15e-3	2.88e-2	1.35e-2	0	0	0	1	0	1

Table 3: Noise robustness under additive Gaussian perturbations. Each cell reports the measured value, while the background encodes its severity tier. The left block gives coefficient error

\varepsilon

on a logarithmic scale; the right block gives structural error (MT

+

ET), which remains zero through 1% noise for all equations, with the first failure appearing for PDE-Divide at 2%.

Axis	Best setting	Portability	Limit
VLM	KIMI-K2.5 gives the top NLS evidence score (4.8/5).	Gemma-4-31B [ 35 ] scores 4.7/5; Qwen-122B/397B [ 36 ] are competitive.	Near/sub-12B VLMs add shock, transport, or potential cues in this test.
LLM	GLM-5.1: rank-1 exact NLS, broadest true-term recall (9/10).	6/8 tested LLMs recover exact NLS at rank 1.	Weaker or smaller LLMs need extra iterations or produce cluttered candidate pools here.
Pipeline	KIMI-K2.5 VLM + GLM-5.1 LLM is the cleanest tested pair.	Several mid/large alternatives also succeed with Agent 4 filtering.	Evidence for sub-12B backends is limited to easier tested cases.

Table 4: LLM/VLM sensitivity analysis on the NLS benchmark: best observed configurations, portability ranges, and practical limits for the VLM and LLM components. Full cue-level rankings refer Appendix.

Configuration	Best selected PDE	$R^{2}$	Reward	NRMSE	Confidence	Final status
Agent 2 enabled	$u_{t}=-uu_{x}-u_{xx}-u_{xxxx}$	1.000000	0.856864	0.000000	99.999977%	SUCCESS
Agent 2 disabled	$u_{tt}=0.073571u_{xx}+0.000938u_{xxx}$	0.252858	0.230022	0.864374	13.56%	EXHAUSTED

Table 5: End-to-end ablation of Agent 2 on the zero noise KS dataset.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Figure A.1: Detailed schematic outlining the working principle of MAGE, the proposed agentic framework for equation discovery.

Panel A: VLM evidence quality for NLS
VLM evidence source	Rank	Score	Correct NLS cues recovered	Misleading cues	Component interpretation
KIMI-K2.5	1	4.8/5	Breathing bound-state pattern, closed peak tracks, linear $k$ – $\omega$ branches, reversible spectral exchange, entropy oscillations, and no shocks.	Mild time-varying-coupling cue.	Richest evidence for smooth conservative NLS dynamics, with a small overinterpretation risk.
Gemma-4-31B	2	4.7/5	Smooth symmetric peaks, closed tracks, tight $k$ – $\omega$ dispersion, periodic $E(k,t)$ and entropy.	None material.	Cleanest evidence for a smooth dispersive breather.
Qwen-3.5-397B	3	4.5/5	Localized breathing, low- $k$ spectral modulation, periodic entropy, explicit “nonlinear balance” cue.	Hints at localized/variable dynamics and clipping.	Strong NLS description; slight risk of trap/potential terms.
Qwen-3.5-122B	4	4.0/5	High-amplitude temporal events, diagonal dispersion branches, low- $k$ energy bursts, entropy inversely tied to coherent states.	Suggests potential wells and time-dependent forcing.	Good phenomenology, slightly less parsimonious than Qwen-397B.
Nemotron-Nano-12B-VL [ 37 ]	5	3.0/5	Recognizes cross-shaped dispersion and periodic spectral entropy.	Missing a global mechanism summary; emphasizes transient/phase-velocity cues.	Useful spectral evidence but incomplete as a synthesis prior.

Appendix

Table B.1: Component-wise NLS ablation: VLM evidence quality.

Panel B: LLM final-candidate audit with fixed KIMI-K2.5 VLM evidence
LLM	Iter.	Best PDE	True-term candidates	Overcomplete among true-term candidates	Dominant spurious terms or residual risk
DeepSeek-V4-Flash [ 39 ]	1	Exact	5/10	4/5	Rank-1 exact; extras add $u_{x}$ , $x^{2}u$ , or $u$ with near-zero coefficients.
DeepSeek-V4-Pro	1	Exact	4/10	3/4	Rank-1 exact; extras include $u$ , $x^{2}u$ , and $\|u\|^{4}u$ .
GLM-5.1	1	Exact	9/10	8/9	Rank-1 exact, but the pool is broad: $u$ , $x^{2}u$ , $xu$ , $u_{xxxx}$ , $\|u\|^{4}u$ , $\|u\|^{2}u_{x}$ , $\|u\|^{2}u_{xx}$ , and derivative-NLS variants remain.
Minimax-M2.7 [ 40 ]	1	Exact	6/10	4/6	Two clean equivalent forms; extras add $u$ , $u_{x}$ , $u_{xxxx}$ , or $\|u\|^{4}u$ .
GPT-OSS [ 41 ]	1	Exact	8/10	6/8	Broad true-term recall, but many overcomplete variants: $xu_{x}$ , $u_{xxxx}$ , $t^{-1}u$ , $\|u\|^{4}u$ , $\|u\|^{2}u_{x}$ , and $\|u\|^{2}u_{xx}$ .

Appendix

Table B.2: Component-wise NLS ablation: final LLM candidate audit using fixed KIMI-K2.5 visual evidence.

Variant	Run	$R^{2}$	Confidence	Final best PDE
With Agent 2	1	0.999880	98.90	$u_{t}=-1.000255uu_{x}-0.999451u_{xx}-0.999187u_{xxxx}$
With Agent 2	2	0.999056	96.93	$u_{t}=-1.008532uu_{x}-1.042205u_{xx}-1.094190u_{xxxx}$
With Agent 2	3	1.000000	100.00	$u_{t}=-1.000000uu_{x}-1.000000u_{xx}-1.000001u_{xxxx}$
With Agent 2	4	1.000000	100.00	$u_{t}=-1.0000uu_{x}-1.0000u_{xx}-1.0000u_{xxxx}$
With Agent 2	5	0.999999	99.90	$u_{t}+0.999975uu_{x}+1.000170u_{xx}+1.000135u_{xxxx}=0$
Without Agent 2	1	0.252858	13.56	$u_{tt}=0.073571u_{xx}+0.000938u_{xxx}$

Appendix

Table B.3: Run-level final outcomes from the KS ablation logs.

Noise	FD verdict	FD $R^{2}$	FD reward	FD conf.	WF verdict	WF $R^{2}$	WF reward	WF conf.
$0\%$	Accept	$0.999940$	$0.909637$	$99.31$	Accept	$1.000000$	$0.909691$	$100.00$
$1\%$	Reject	$0.439340$	$0.399663$	$25.12$	Accept	$0.999836$	$0.909542$	$98.72$
$2\%$	Reject	$0.668092$	$0.572464$	$42.39$	Accept	$0.999909$	$0.856786$	$99.05$
$5\%$	Reject	$-0.087811$	$-0.071951$	$51.61$	Accept	$0.999990$	$0.909681$	$99.68$
$10\%$	Reject	$0.527865$	$0.452308$	$31.29$	Accept	$0.999945$	$0.909641$	$99.26$
Mean	–	$0.5095$	$0.4524$	$49.94$	–	$0.9999$	$0.8991$	$99.34$

Appendix

Table B.4: Final validation metrics reported by the agentic discovery logs. The acceptance threshold is

80\%

confidence. FD reaches the threshold only for the clean

0\%

case, while WF succeeds at every noise level.

Noise	FD final best PDE	WF final best PDE
$0\%$	$u_{t}=(0.5i)u_{xx}+i\|u\|^{2}u$	$u_{t}=(0.500000i)u_{xx}+(1.000000i)\|u\|^{2}u$
$1\%$	$u_{tt}=5.5753u_{xx}-0.2364x^{4}u$	$u_{t}=(-0.001409+0.498327i)u_{xx}+(-0.000571+1.000553i)\|u\|^{2}u$
$2\%$	$u_{t}=(0.005518+0.000003i)i\,u_{xx}+(0.210816-0.020704i)i\,\|u\|^{2}u+(3.508637+0.090500i)i\,e^{-x^{2}}u$	$u_{t}=(-0.008381+0.503541i)u_{xx}+(-0.001953+1.000706i)\|u\|^{2}u+(-0.000123+0.000055i)u_{xxxx}$
$5\%$	$u_{tt}=(-67.7770+0.1003i)u+(18.8292-0.0013i)\|u\|^{2}u+(18.1959-0.0006i)u_{xx}+(0.0054)u_{xxxx}$	$u_{t}=(-1.14{\times}10^{-4}+0.500661i)u_{xx}+(1.55{\times}10^{-4}+0.999476i)\|u\|^{2}u$
$10\%$	$u_{tt}=(18.9627+0.0016i)u_{xx}+(0.005736+10^{-6}i)u_{xxxx}+(10.4994+0.6554i)\|u\|^{2}u$	$u_{t}=(0.000056+0.498982i)u_{xx}+(-0.000087+0.997236i)\|u\|^{2}u$

Appendix

Table B.5: Final best PDEs from the response logs. The WF equations remain close to the target NLS form in Eq. ( B.3 ). FD recovers the correct NLS equation in the clean case, but drifts to non-NLS structures when noise is present.

Figure B.1: Agent 1 diagnostic output for the KS dataset. Eight deterministic panels form the evidence object passed to Agent 2. Top: (1) field contour $u(x,t)$ ; (2) spatial derivative $u_{x}$ ; (3) temporal derivative $u_{t}$ ; (4) spatial spectrum $E(k)$ at five snapshots showing the active band and high-wavenumber rolloff. Bottom: (5) local-extrema trajectories; (6) $k$ – $\omega$ dispersion spectrum; (7) spectral energy evolution $E(k,t)$ ; (8) normalized spectral entropy $H(t)$ , saturating near $H\!\approx\!0.37$ .

Noise	$\hat{c}_{xx}$	Rel. err.	$\hat{c}_{\mathrm{nl}}$	Rel. err.	Extra selected term
$0\%$	$0.500000i$	$0.000$	$1.000000i$	$0.000$	none
$1\%$	$-1.409\mathrm{e}{-3}+0.498327i$	$0.437$	$-5.71\mathrm{e}{-4}+1.000553i$	$0.079$	none
$2\%$	$-8.381\mathrm{e}{-3}+0.503541i$	$1.820$	$-1.953\mathrm{e}{-3}+1.000706i$	$0.208$	$\|\hat{c}_{xxxx}\|=1.35\mathrm{e}{-4}$
$5\%$	$-1.14\mathrm{e}{-4}+0.500661i$	$0.134$	$1.55\mathrm{e}{-4}+0.999476i$	$0.055$	none
$10\%$	$5.60\mathrm{e}{-5}+0.498982i$	$0.204$	$-8.70\mathrm{e}{-5}+0.997236i$	$0.277$	none

Appendix

Table B.6: Weak-form coefficient stability relative to

c_{xx}^{\star}=0.5i

and

c_{\mathrm{nl}}^{\star}=i

. Relative errors are reported in percent.

Noise	$\|u\|$ range	$u_{x}$ range	$\max\|u_{x}\|$	$u_{t}$ range	$\max\|u_{t}\|$
$0\%$	$[0.01599,\,4.00729]$	$[-9.265,\,9.265]$	$9.27$	$[-4.994,\,4.993]$	$4.99$
$1\%$	$[0.00991,\,4.01648]$	$[-9.556,\,9.791]$	$9.79$	$[-6.865,\,7.452]$	$7.45$
$2\%$	$[0.00300,\,4.01147]$	$[-10.26,\,10.24]$	$10.26$	$[-12.72,\,15.44]$	$15.44$
$5\%$	$[0.00039,\,4.09216]$	$[-11.71,\,13.26]$	$13.26$	$[-37.24,\,35.81]$	$37.24$
$10\%$	$[0.00030,\,4.12026]$	$[-23.14,\,16.21]$	$23.14$	$[-77.46,\,79.33]$	$79.33$

Appendix

Table B.7: Noise amplification in finite-difference derivative panels. Even moderate additive noise produces large derivative excursions, particularly for

u_{t}

Removed evidence	Rounds	Status	Confidence	Best validated PDE
None (full panel)	1	Success	100.00%	$u_{t}=-uu_{x}-u_{xx}-u_{xxxx}$
$u(x,t)$ contour	3	Exhausted	55.26%	$u_{t}=-0.089838uu_{x}-1.260220u_{xxt}$
$H(t)$ entropy	3	Exhausted	49.88%	$u_{tt}=0.0335u_{xx}-0.8701u_{ttxx}+0.0233u_{xxxx}$
$k$ – $\omega$ spectrum	3	Exhausted	52.27%	$u_{t}=0.0024u^{2}u_{x}-0.8778u_{xxt}$
Extrema trajectories	3	Exhausted	45.96%	$u_{t}=-0.661015uu_{x}-0.268456u_{xx}+0.010440u^{3}$
$u_{x}(x,t)$ contour	3	Exhausted	46.77%	$u_{t}=-0.848078u_{xxt}-0.027353uu_{x}+0.000143u_{xxxxx}$

Appendix

Table B.8: Single-diagnostic removal ablation of the Agent 1 visual evidence panel for KS discovery. Only the complete eight-plot panel produced an accepted PDE; every ablated run exhausted the inner loop after three synthesizer rounds.

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Discovering PDEs in heterogeneous media requires jointly identifying the governing operator and the unknown spatial fields that parameterize it. These tasks are coupled: changing field placement changes the differential law, while a sufficiently flexible field can conceal structural error on a single trajectory. We present Hypothesize, Evaluate, Refine for PDE Discovery (HER-PDE), a scientific-agent framework that discovers compositional PDE structure together with nonparametric, time-invariant coefficient fields. The Agent analyzes two noisy trajectories generated by different excitations, proposes complete expression-tree hypotheses, and combines creative structural exploration with local candidate refinement. Its Hypothesis Evaluation Interface (HEI) estimates only the fields explicitly declared in each hypothesis, never adds missing terms, and scores structures by bidirectional cross-excitation transfer. The selected law is subsequently audited on a sealed temporal interval. Across five controlled two-dimensional systems observed with 5 percent relative Gaussian state noise, the Agent recovers the generating operator in all five cases, including equivalent signed-field and product-rule parameterizations. Across nine unknown coefficient fields, the recovered fields attain a median Pearson correlation of approximately 0.85 and a median relative L2 error of approximately 0.28. These results show that agent-guided hypothesis refinement can recover heterogeneous governing laws without prescribing a parametric form for their spatial coefficients.

Abstract

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