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

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

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

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

arXiv:2608.27869v1Aug 28, 2026cs.AI

TL;DR

PDE discovery requires inferring governing laws from observational data despite constraints from predefined libraries, noise, hallucination, and limited iterative refinement. MAGE addresses this with a multimodal, role-specialized hypothesis-validation loop, achieving exact recovery across the evaluated canonical suite and strong coefficient accuracy, while broader transfer remains unvalidated.

Problem
PDE discovery from observational data is limited by predefined libraries, noise sensitivity, hallucination, and insufficient iterative interaction with observational evidence.
Method
MAGE uses a hierarchy of conditioned stages and four specialized agents to extract diagnostics, infer physical abstractions, synthesize library-free candidate PDEs, and numerically validate them.
Results
MAGE achieves 100.0% exact structural recovery on the evaluated canonical suite, with the strongest coefficient fidelity on seven of eight equations and approximately 3 orders of magnitude lower geometric-mean coefficient error.
Takeaways & Limitations
The findings provide scoped evidence for iterative agentic governing-law discovery across canonical PDEs, complex geometries, and a laboratory sensor record.
Takeaways & Limitations
Broader real-world transfer remains uncertain because evaluation covers one sensor record and the held-out record participates in candidate ranking.

Abstract

from arXiv · show

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 \textbf{MAGE} (\textbf{M}ultimodal \textbf{A}gentic \textbf{G}overning \textbf{E}quation Discovery), an agentic framework that organizes PDE discovery as a \textit{confidence governed hypothesis validation loop} inspired by the scientific cycle of observation, hypothesis, and falsification. Four role-specialized agents collaborate: a \textit{Differential Observer} computing derivatives and diagnostic visualizations; a VLM-powered \textit{Phenomenology Extractor} distilling qualitative cues from multimodal diagnostics; an LLM-driven \textit{Governing Law Synthesizer} proposing candidates without a predefined library; and an \textit{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 \textbf{8/8} exact structural recovery and the lowest coefficient error among the compared methods on \textbf{7/8} systems, with improvements of up to \textbf{4 orders of magnitude} and a geometric-mean improvement of approximately \textbf{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.

1 Introduction

MAGE frames PDE discovery as an iterative, confidence-governed cycle that combines multimodal observation, hypothesis generation, and numerical validation. Its role-specialized agents provide structured evidence and an explicit accept-reject protocol without a predefined symbolic library.

Existing PDE discovery methods face distinct constraints from fixed libraries, combinatorial symbolic search, hallucination, prompt sensitivity, or limited task specialization.
MAGE operationalizes discovery as a confidence-governed hypothesis-validation loop inspired by observation, hypothesis, and falsification.
The framework extracts numerical diagnostics, infers physical abstractions, synthesizes candidate equations, and validates them quantitatively in successive stages.
Visual evidence such as contours, gradients, and spectra is converted into qualitative regime descriptors that condition candidate-equation proposals.
Candidates are rejected and regenerated when confidence falls below a user-specified threshold, making numerical evidence determine the stopping criterion.

2 Related Work

Related work spans library-based, symbolic-regression, LLM-based, and agentic approaches to governing-equation discovery. These methods trade fixed expressivity, search cost, model dependence, and physical priors in different ways.

Library-based methods perform sparse regression over predefined candidate terms, so their expressivity depends on the included library.
Symbolic-regression methods search open-form expressions through genetic programming, reinforcement learning, or constrained grammars, with complexity and stabilization trade-offs.
LLM-based methods use pretrained language models to generate or refine mathematical structures through libraries, templates, evolutionary search, or hierarchical prompting.
Agentic frameworks add iterative reasoning and tool use, including code execution or physics-informed priors such as symmetries and conservation laws.
Existing agentic demonstrations have mainly been evaluated on algebraic or low-dimensional forms rather than broader PDE settings.

3 Methodology

MAGE decomposes PDE discovery into conditioned latent inference stages that transform observations into physical abstractions, symbolic candidates, and numerical validation metrics. Specialized agents implement this workflow using diagnostic computation, VLM interpretation, LLM synthesis, and executable scoring.

3.1 Problem Formulation: Direct optimization over symbolic operators and coefficients is computationally intractable because candidate operators grow combinatorially while coefficients require continuous optimization.
3.2 Agentic Latent Variable Decomposition: The latent-variable formulation maps raw observations through numerical diagnostics, semantic physical abstractions, executable candidate programs, and evaluation metrics.
3.2 Agentic Latent Variable Decomposition: MAGE assigns successive inference factors to role-specialized agents that perform observation, abstraction, hypothesis generation, and falsification.
3.3 Agent Realization: The Differential Observer computes derivatives, contours, spectra, extrema trajectories, dispersion features, spectral-energy evolution, and entropy for time-dependent fields.
3.3.1 Wavelet Denoising: For structured grids, noisy fields are denoised with a 2D sym8 wavelet transform, while complex-field components are perturbed independently.
3.3 Agent Realization: The VLM converts diagnostic visual evidence and metadata into structured physical-regime descriptors without naming PDEs or generating symbolic expressions.
3.3 Agent Realization: The LLM synthesizes candidate equations de novo from semantic conditioning and numerical data, without assuming a fixed candidate library.
3.3 Agent Realization: Candidates are evaluated with normalized ordinary least squares on a deterministic 80/20 split and scored for numerical consistency and parsimony.

4 Results and Discussion

MAGE is evaluated across canonical PDE benchmarks, experimental sensor data, complex geometries, noise levels, foundation models, and diagnostic ablations. It achieves complete structural recovery on the canonical suite while showing useful but bounded performance under experimental, noisy, and component-variation settings.

4.1 Canonical PDE Benchmark Suite: MAGE obtains 100.0% exact recovery with zero structural failures on the evaluated canonical PDE suite.The comparison reports structural recovery alongside coefficient estimation accuracy.
4.1 Canonical PDE Benchmark Suite: Approximately 3 orders of magnitude lower geometric-mean coefficient error and strongest coefficient fidelity on seven of eight equations distinguish MAGE from the compared methods.LLM-based baselines reach 85.7% and 71.4% exact recovery, respectively, while residual errors remain in stiff and coupled cases.
4.2 Experimental Sensor Data: R^2 = 0.98538, NRMSE = 0.12090, and confidence = 87.91% accompany selection of the cubic model on the Silverbox sensor record.The closest linear alternative has NRMSE = 0.12684 and confidence = 87.32%, while held-out error ranking retains the cubic term.
4.3 Complex Geometric Domains: MAGE recovers Laplacian support with zero missing or spurious terms and low single digit coefficient error on a 2D plate and a 3D space shuttle.These complex-geometry cases address steady elliptic discovery from spatial contour observations.
4.5 Noise Robustness: Structural error remains zero through 1% noise for all equations, while coefficient error generally increases and PDE-Divide fails first at 2% noise.Several smooth transport/diffusion equations retain comparatively small coefficient error through 5–10% noise, whereas PDE-Divide and Conv.-Diff. degrade earlier.
4.6 Ablation Study: Multiple foundation-model backbones recover the NLS target structure, but their roles differ: KIMI-K2.5 reaches a VLM score of 4.8/5, while GLM-5.1 gives the most reliable rank-1 recovery.The analysis reduces concern that the NLS result depends on one backbone, without establishing model independence across PDE families.
4.6 Ablation Study: Removing Agent 2 or any single diagnostic panel prevents acceptance on KS, with spectral-panel removal lowering confidence to 6.92% or 7.16%.The full panel accepts the correct KS structure in the first outer iteration with 100% confidence; local panels show milder 20–55% degradation when removed.

5 Conclusion

MAGE frames PDE discovery as a confidence-governed validation loop that separates observation, physical abstraction, symbolic synthesis, and numerical falsification. The evaluated findings provide scoped evidence for iterative agentic discovery, while broader transfer remains open.

MAGE reformulates PDE discovery as a confidence-driven hypothesis-validation loop with explicit separation of perception, physical abstraction, symbolic synthesis, and numerical falsification.
MAGE identifies target structures across the canonical suite, lowers aggregate coefficient error, finds Laplacian support in two complex geometries, and selects the expected cubic nonlinearity in one sensor record.
The results constitute scoped evidence for iterative agentic governing-law discovery rather than universal generalization.
Evaluation is constrained by off-the-shelf pretrained backbones, noisy-observation bottlenecks, one sensor record, one system class, and a held-out record used in candidate ranking.
The pipeline transforms raw observations into validated equations through sequential stages of observation, phenomenological abstraction, hypothesis generation, and numerical falsification.
If no candidate clears the confidence threshold, inner refinement regenerates candidates and outer restarts force fresh observational interpretation.

A.2 Compute Resources and API-Based Inference

MAGE runs orchestration and validation locally on a standard workstation while outsourcing model inference to serverless foundation-model endpoints. This design avoids local GPU or HPC requirements and uses a predefined library only for baseline comparisons.

Experiments used a standard workstation with 6 CPU cores and 32 GB RAM for orchestration, parsing, validation execution, and figure generation.
Model inference did not require local GPU acceleration or high-performance computing clusters.
NVIDIA-hosted serverless endpoints supplied inference through moonshotai/kimi-k2.5 for visual reasoning and z-ai/glm-5.1 for language, discovery, and validation tasks.
The local pipeline routes multi-agent prompting, candidate formulation, code execution for validation, and self-correction loops.
Library-based baselines were compared using one consistent over-complete predefined symbolic library from which sparse-regression methods selected governing terms.

B.2 Extended VLM–LLM Component Ablation on NLS

The NLS component ablation separates the roles of visual evidence quality and language-model candidate selection. Stronger visual interpretation better constrains the physical search, while LLMs mainly differ in candidate-set parsimony and refinement behavior.

VLM evidence ranking: KIMI-K2.5 provides the richest NLS description, capturing localized breathing, dispersive k–ω branches, reversible spectral exchange, entropy oscillations, and absent shocks.
VLM evidence ranking: Qwen VLMs introduce potential or variable-coefficient interpretations, while Nemotron and Phi-4 add false shock, smoothing, or transport cues that can seed unsuitable candidates.
LLM candidate-set behavior: Every tested LLM eventually places an exact NLS form at rank 1 when conditioned on the same KIMI-K2.5 evidence.
LLM candidate-set behavior: DeepSeek-Flash, DeepSeek-Pro, and Minimax recover the correct equation in one outer iteration with relatively simple alternatives, whereas other models retain extra operators or wrong-target candidates.
LLM candidate-set behavior: The ablation assigns VLM quality to physical well-posedness of the search prior and LLM quality to parsimony of the symbolic candidate pool.

B.3 Extended Ablation on Agent 2: Phenomenology Extractor

The Agent 2 ablation tests whether phenomenological abstraction improves PDE discovery on a zero-noise Kuramoto–Sivashinsky dataset. Enabling it produces accurate first-iteration recovery, whereas removing it yields a rejected low-confidence surrogate.

The ablation compares Agent 2 enabled versus disabled on the same zero-noise one-dimensional spatio-temporal Kuramoto–Sivashinsky dataset.
With Agent 2, MAGE recovers the correct KS equation in the first outer iteration with R2 = 1.000000, NRMSE = 0.000000, and Confidence = 99.999977%.
Without Agent 2, the system exhausts its inner loop and selects a wave-like surrogate with R2 = 0.252858, NRMSE = 0.864374, and Confidence = 13.56%.
Enabling Agent 2 raises selected reward by approximately 3.7×, increases confidence by about 86.4 percentage points, and changes the verdict from REJECT_PDE to ACCEPT_PDE.
Successful Agent 2 runs recover coefficients near the expected unit magnitudes, including −1.000255, −0.999451, and −0.999187 in one run.

B.3.2 Failure Mode Without Agent 2

Without Agent 2, the system misses the correct first-order KS candidate class and repeatedly selects low-confidence wave-like surrogates. Visual phenomenology is therefore essential for steering hypothesis generation toward the specific KS balance of terms.

The disabled run omits the correct first-order KS structure combining u_xx, u_xxxx, and u u_x, despite generating some of its individual terms.
12.01% confidence and R2 = 0.225745 characterize the first disabled selection, u_tt = 0.065263u_xx.
Retrying candidate generation does not fix the structural error; the final disabled model remains a low-confidence second-time-derivative surrogate.
Agent 2 resolves misleading advection, dispersive, and temporal-correlation cues into the KS balance of nonlinear advection, second-order instability, and fourth-order stabilization.
With Agent 2, the correct equation enters the candidate set before validation; without it, validation must choose among misspecified candidates.
The ablation identifies the Contour Plot Analyser as necessary for KS discovery and supports visual and spectral reasoning as active hypothesis-generation evidence.
Because the ablation changes support placement and candidate lists as well as disabling Agent 2, fixed-validator replication remains an additional useful diagnostic.

B.4 Finite-Difference Versus Weak-Form Validation

This appendix compares finite-difference and weak-form validation for noisy NLS observations, showing why MAGE favors weak-form evaluation. Weak forms transfer derivatives onto test functions, improving robustness while finite differences amplify noise and can change the identified model class.

B.4 Finite-Difference Versus Weak-Form Validation: Weak-form validation fits integrated identities against compactly supported test functions, whereas finite differences fit pointwise identities using numerically differentiated data.
B.4 Finite-Difference Versus Weak-Form Validation: The evaluated target is the focusing cubic NLS on a shared space-time grid, with the target recovered by clean FD and weak-form runs.
B.4 Finite-Difference Versus Weak-Form Validation: At the 80% confidence threshold, FD succeeds only at 0% noise, while WF succeeds at every tested noise level.
B.4 Finite-Difference Versus Weak-Form Validation: WF remains near the target NLS family, while FD shifts under noise toward wave-like, stiff-wave, and higher-order nonlinear structures.
B.4.2 Why Weak-Form Validation Remains Stable?: Weak-form validation transfers derivatives from noisy fields onto smooth test functions, enabling local averaging and preserving coefficient-fitting regression structure.
B.4.2 Why Weak-Form Validation Remains Stable?: The diagnostic panels include field, derivative, spectral, extrema-trajectory, dispersion, spectral-energy, and entropy views of the KS dataset.
B.4.2 Why Weak-Form Validation Remains Stable?: Across noise levels, weak-form refinement stays within the cubic NLS family, with coefficient deviations remaining small and a 1.35 × 10^-4 fourth-derivative correction at 2% noise.
B.4.2 Why Weak-Form Validation Remains Stable?: WF preserves sparse structure and physically correct coefficient scales, while finite-difference derivative ranges expand sharply under noise, especially for u_t.

B.4.1 Main Quantitative Result

Weak-form validation is substantially more robust than finite differences across the NLS noise sweep. It preserves the correct cubic-NLS structure and coefficient fidelity, whereas finite differences become structurally fragile once noise is introduced.

WF accepts the correct NLS model at all five noise levels, while FD accepts it only at 0% noise and fails at every nonzero level.
99.34% mean confidence, 0.8991 mean reward, and 0.9999 mean test R2 are achieved by WF across the five runs.
WF remains in the cubic-NLS family with coefficients close to (c_xx, c_nl) = (0.5i, i), while FD changes model class under noise.

B.5 Ablation Study on Agent 1 Outputs

The ablation study shows that the complete eight-view diagnostic panel is necessary for reliable KS discovery. Local views anchor field geometry, while spectral views provide the strongest evidence for derivative orders and stabilization.

Diagnostic panel: The full eight-panel diagnostic evidence combines field, derivative, trajectory, spatial-spectral, k–ω, spectral-energy, and entropy views of one spatiotemporal field.These complementary views jointly expose local geometry, temporal behavior, modal structure, energy transfer, and entropy evolution.
Ablation outcome: R^2 = 1.0000, reward 0.8569, and confidence 100.00% were achieved with all plots retained, while every single-diagnostic removal failed the 80% threshold.The full panel was accepted in one outer iteration; each ablation exhausted the inner loop after three synthesizer rounds and two rejection-router interventions.
Mechanistic evidence: The full panel identifies nonlinear steepening, low-to-high wavenumber transfer, entropy saturation, and bounded oscillations supporting the KS balance.The resulting candidate library spans nonlinear, second-order, higher-order, and mixed derivative terms before the KS-consistent candidate appears.
Local panels: Removing local panels disrupts field geometry and propagation cues, causing searches to drift toward alternative regularized-wave models.The local views encode amplitude, phase organization, coherent morphology, gradient sharpening, and temporal behavior.
Spectral panels: Removing spectral panels weakens mechanistic discrimination: E(k) confidence falls to 6.92%, and the two spectral outcomes lie below the 45–55% local-panel range.The spectral views reveal active scales, temporal cascades, and high-wavenumber suppression associated with the fourth-order stabilizing term.
Interpretation: Only the joint spectral signature together with local geometry suppresses KdV/Kawahara, Burgers, and BBM false positives and identifies the correct KS derivative orders.The full panel enables generation and acceptance of the correct candidate in the first outer iteration.

C Running Example of Multimodal Evidence Construction

The running example constructs multimodal evidence from a structured complex x–t field, combining differential, geometric, and spectral diagnostics. Weak-form validation then accepts a parsimonious two-term governing equation with perfect fit metrics and 100.00% confidence.

Evidence construction: The pipeline begins from a loaded structured-grid payload and generates deterministic diagnostic visual evidence for downstream agents.The Differential Observer is instructed to characterize variables, axes, reliability, panels, geometry, structure, mechanisms, coefficient variation, and further diagnostic requests.
Diagnostic views: The evidence panel includes |u|, spatial and temporal derivatives, extrema tracks, E(k), the k–ω spectrum, E(k,t), and normalized entropy H(t).The panels are designed to expose geometry, rates of change, modal structure, energy evolution, and entropy.
Data characterization: The input is a two-dimensional structured x–t grid with Nx = 512, Nt = 501, uniform sampling, and a complex field displayed through |u|.The reported spacings are dx = 0.01953 and dt = 0.006283.
Spectral observations: The extracted phenomenology reports steep high-k decay, diagonal k–ω branches with slopes approximately ±4–±5, periodic low-k concentration, and redistribution toward higher modes.These observations accompany strong energy near ω = 0 and characterize the field’s spectral evolution.
Phenomenology: The field exhibits periodic recurrence with Δt ≈ 1.57, stationary spatial features, no visible shocks, reversible scale exchange, and entropy oscillating roughly between 0.05 and 0.35.Its structure is smooth, globally oscillatory, bilaterally symmetric about x = 0, and temporally periodic.
Decision: R^2 = 1.000000, NRMSE = 0.000000, reward 0.909691, and confidence 100.00% lead to VERDICT=ACCEPT_PDE in outer iteration 1.Extra-term variants are penalized for lack of parsimony, while the quintic-only alternative reaches only 52.15% confidence.