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Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fields

YuJie Huang, WenWu He, ZhuoEr Lin, Congcong Liu, Dong Liang, Zhuo-Xu Cui

arXiv:2608.27475v1cs.AIcs.LG

TL;DR

Discovering PDEs with heterogeneous media requires jointly identifying operator structure and unknown spatial coefficient fields. HER-PDE uses complete expression-tree hypotheses, nonparametric field estimation, and bidirectional transfer across two excitations; across five controlled systems, it recovered all generating operators and estimated nine fields with median Pearson correlation of approximately 0.85 and median relative L2 error of approximately 0.28.

  • Problem

    Existing approaches often fix either PDE structure or coefficient representation, leaving compositional search with candidate-specific, nonparametric spatial fields underexplored.

  • Method

    HER-PDE has a scientific Agent propose complete expression-tree hypotheses while HEI preserves each tree, fits only its declared fields, and evaluates bidirectional transfer between two noisy trajectories.

  • Results

    Across five two-dimensional systems with 5% relative Gaussian state noise, the Agent recovered the generating operator structure in all five cases, including equivalent parameterizations.

  • Takeaways & Limitations

    The framework extends PDE discovery to structured laws with nonparametric spatial heterogeneity without prescribing a parametric form for coefficient fields.

  • Takeaways & Limitations

    The experiments establish five positive discovery case studies rather than a population discovery rate, and time-varying, spatiotemporal, and stochastic coefficient laws remain outside the current scope.

Abstract

from arXiv · show

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.

1 Introduction

HER-PDE jointly discovers compositional PDE structure and unknown, time-invariant spatial coefficient fields by evaluating complete hypotheses across two differently excited noisy trajectories.

  • Problem: Structural placement, field identity, spatial dependencies, and field reuse are coupled because changing field placement changes the governing operator.Flexible fields can also compensate for an incorrect operator on a single trajectory.
  • Problem: Existing approaches commonly prescribe either the structural representation or the coefficient representation, leaving compositional search with candidate-specific field estimation underexplored.Each candidate structure defines its own function estimation problem.
  • Framework: Two trajectories share the same operator and time-invariant fields but differ in excitation, enabling bidirectional transfer to test whether fitted fields describe the shared mechanism.Different excitations expose different state and derivative combinations while leaving the fields fixed.
  • Framework: HER-PDE represents a complete expression tree and its unknown coefficient fields as one hypothesis while preserving operator scope, nonlinear operations, spatial dependencies, and field reuse.HEI fits only fields declared by the submitted tree and does not alter its operator structure.
  • Framework: The Agent combines global hypothesis generation, transfer-based revision, field diagnostics, and final selection, followed by auditing on a sealed temporal interval.HEI provides the evaluation scores used throughout this workflow.
  • Results: Five controlled two-dimensional studies with 5% relative Gaussian state noise recover the generating operator, including equivalent signed-field and product-rule parameterizations.The cases include conservative and nonconservative placement and shared and independent fields.

2 Related Work

Related work spans fixed libraries, expandable vocabularies, open-form symbolic grammars, and multiple coefficient regimes; HER-PDE combines open-form structure search with declared nonparametric spatial fields.

  • Taxonomy: Equation-discovery methods vary along structural representations and coefficient regimes, including constants, compact expressions, time-invariant spatial fields, time-varying fields, and irregular or stochastic fields.Figure 1 positions representative works by these dimensions, but within-cell placement has no quantitative meaning.
  • Prior approaches: Closed-library methods select terms from finite candidate matrices, while expandable-library methods enlarge bounded structural vocabularies during learning.Examples include SINDy and PDE-FIND for closed libraries, and PDE-Net 2.0 and DLGA-PDE for expandable libraries.
  • Prior approaches: Open-form grammars construct complete expressions from operands and operators, supporting free-form searches for differential operators beyond fixed term libraries.SGA-PDE, DISCOVER, R-DISCOVER, and EqGPT exemplify this family.
  • HER-PDE: HER-PDE makes coefficient fields part of the compositional hypothesis, keeping c(x)∆u distinct from ∇·(c(x)∇u) and retaining their operator–field structures.HEI estimates the declared fields and induced spatial derivatives after the tree is fixed.
  • Coefficient representations: Classical inverse problems estimate fields for a known structure, whereas equation discovery must select the structure while estimating its coefficient functions.Variable coefficients therefore couple structural selection to function estimation.
  • Coefficient representations: The closest prior regime uses nonparametric time-invariant spatial fields, including spatial kernels, B-splines, and prescribed feature dictionaries.HIN-PDE, WG-IDENT, and related methods differ in structural assumptions and estimation devices.
  • Scope: Time-varying, spatiotemporal, and stochastic coefficient laws remain outside the current scope of HEI’s time-invariant spatial-field candidate class.The current representation uses spatial kernels for time-invariant fields.
  • Scientific agents: HER-PDE extends proposal, numerical evaluation, and revision workflows to heterogeneous PDEs by fitting only declared fields and testing transfer across independent excitations.Field fitting and cross-trajectory evidence guide both revision and final selection.

3 Method

HER-PDE formulates PDE discovery as joint inference of a complete expression tree and unknown spatial coefficient fields. HEI fits only declared fields, evaluates candidates through cross-trajectory transfer, and preserves a sealed temporal interval for final auditing.

  • Problem formulation: The Agent represents each candidate as a complete expression tree specifying terms, nonlinear operations, derivatives, operator scope, and coefficient-field placement or reuse.Each field’s spatial dependency set and the total number of fields are determined by the candidate structure.
  • Problem formulation: The admissible candidate class is nonlinear in measured states and derivatives but linear in unknown fields and their induced spatial derivatives once the tree is fixed.This permits field-bearing operators whose placement determines which field derivatives enter the law.
  • Hypothesis Evaluation Interface: HEI compiles and evaluates the submitted tree without algebraic rearrangement, fitting only the coefficient fields explicitly declared by the hypothesis.Every evaluated term and operator therefore originates from the Agent’s submitted structure.
  • Hypothesis Evaluation Interface: HEI estimates fields jointly in one design matrix after subtracting any field-free candidate component from the time derivative.The resulting fit uses frozen derivatives reconstructed from noisy observations with separable tensor-product Savitzky–Golay operators.
  • Candidate evaluation: Search scores measure bidirectional transfer between source and target trajectory intervals, with lower values indicating better transfer.Symmetric transfer favors fields explaining both experiments rather than flexible functions interpolating one trajectory.
  • Candidate evaluation: After selection, HEI refits using the first three temporal intervals and evaluates the other trajectory’s final interval as a sealed audit.The final 20% is unavailable to candidate generation, search scoring, and hyperparameter selection.
  • Scientific-agent workflow: The workflow combines open-ended analysis with global exploration and local refinement, while enforcing an initial archive of competing scored candidates before the search proceeds.The controller checkpoint converts early data analysis into explicit competing explanations and structured numerical revision.

4 Experiments

Across five controlled two-dimensional systems, HER-PDE recovered the generating operator under 5% noisy observations, while also recovering nine unknown coefficient fields. Field estimates were compared at shared anchors, and the study reports five positive case studies rather than a population discovery rate.

  • Experimental protocol: Five two-dimensional systems were evaluated using raw coordinates, times, and state measurements with 5% relative Gaussian noise.Clean states, simulator derivatives, generating equations, and oracle candidates remained hidden from the Agent.
  • Governing-structure discovery: The Agent submitted the generating structure in all five registered cases.The cases include variable diffusion, reaction–diffusion, advection–diffusion, groundwater flow, and orthotropic diffusion.
  • Governing-structure discovery: The search retained first generating-structure candidates while continuing to test alternative operator placements, dependencies, and extra terms.Several later proposals were invalid, and the trajectory was not a monotone numerical optimization.
  • Governing-structure discovery: Groundwater’s lowest search error was 0.51318, whereas the generating conservative operator scored 0.51398 and had the slightly smaller sealed-window error, 0.122599 versus 0.122641.The Agent selected the conservative operator using transfer, operator coherence, and unsupported-complexity considerations rather than the smallest search-visible number.
  • Unknown-field recovery: Across nine coefficient fields, Pearson correlation ranged from 0.711 to 0.966 and relative L2 error ranged from 0.082 to 0.736.The largest amplitude error was for the advection field −vx, with Erel = 0.736 and r = 0.740.
  • Evidence provenance: The experiments establish five positive discovery case studies and do not estimate a population discovery rate.Population-level rates require additional unseen field realizations, preparations, noise seeds, and Agent samples.

5 Conclusion

HER-PDE combines complete symbolic hypotheses with nonparametric coefficient-field estimation and cross-trajectory evidence. Across five noisy two-dimensional systems, it recovered the generating operator structures while addressing heterogeneous fields and competing operator placements.

  • Conclusion: HER-PDE discovers PDE structures whose coefficients are unknown spatial fields.Its Agent constructs complete symbolic hypotheses while preserving operator scope, spatial dependencies, and shared-field identity.
  • Conclusion: HEI estimates declared coefficient fields nonparametrically and measures transfer between independent experiments.The workflow combines open-ended analysis, creative hypothesis generation, and local candidate refinement.
  • Conclusion: Across five two-dimensional systems observed with 5% state noise, the Agent recovered the generating operator structure in every case.The systems include groundwater flow, variable diffusion, reaction–diffusion, advection–diffusion, and orthotropic diffusion.
  • Conclusion: The cases distinguish conservative and nonconservative placement, shared and independent fields, and diffusion, transport, and reaction mechanisms.The decision traces show that structural coherence was used alongside HEI scores when selecting among evaluated hypotheses.

A Agent Decision Traces

The appendix condenses chronological Agent conversations and HEI candidate ledgers into auditable decision traces. It preserves scored equations, errors, validation outcomes, and the Agent’s contemporaneous rationale.

  • A Agent Decision Traces: Scored equations and errors are copied from candidate ledgers, while validation outcomes come from chronological conversations.The update column paraphrases the Agent’s contemporaneous rationale.
  • A Agent Decision Traces: The appendix writes ∆u = uxx + uyy and suppresses unrestricted two-dimensional field arguments unless stated otherwise.Under this convention, Cj = Cj(x, y), and lower Esearch is better.

A.1 Detailed trajectory: advection–diffusion

The advection–diffusion trajectory shows HER-PDE combining structural exploration with falsification through HEI transfer scores, selecting diffusion plus two independent transport fields.

  • Phase I reasoning: Phase I diagnostics briefly favored a wave-like account, but HEI rejected it because the candidate language permits only a first-order time rate on the left.The exploratory correlation between utt and ∆u was stronger than that between ut and ∆u.
  • Candidate refinement: Eval. 31 achieved a score of 0.6588 with diffusion, two transport fields, and reaction, while Eval. 37 improved transfer to 0.4944 after removing reaction.The sequence tested whether reaction was necessary after introducing a joint diffusion–transport construction.
  • Candidate refinement: A nonlinear reaction extension scored 0.6258, and directional ablations scored 1.7138 and 1.6633, showing that both independent transport components were required.The ablations removed one transport direction at a time.
  • Final selection: The final archive winner was ut = C0∆u + C1ux + C2uy with search score 0.4944 and sealed error 0.5114.It followed component, nonlinearity, dependency, and operator-placement controls.
  • Final selection: The Agent first composed diffusion with two independent spatial transport fields, then tested reaction, field coupling, and operator placement as falsification alternatives.Deleting a transport direction, coupling directions through one field, moving diffusivity inside divergence, and splitting diffusion reduced transfer performance.

A.2 Compressed traces for all five discoveries

Across five discoveries, the controller used immediate falsification, composition of incomplete operator families, and changes in operator placement or shared-field identity.

  • Compressed traces: Table 4 summarizes each positive case through its decisive structural transition and final selection controls.The table concerns the five registered discoveries.
  • Compressed traces: The five trajectories exhibit three discovery patterns: immediate proposal followed by attempted falsification, composition of incomplete operator families, and discrete structural changes.These patterns describe the controller’s behavior across the completed archives.
  • Compressed traces: HEI provides comparable numerical evidence for each transition, while the Agent chooses which structural alternative to construct and which scored law to submit.The controller combines numerical evaluation with Agent-directed structural selection.

B Top-Five HEI Candidate Tables

The candidate tables rank structurally distinct hypotheses by search-visible cross-trajectory error and document their operator equivalence, field dependencies, and representative alternatives.

  • Ranking and equivalence: Candidates are ordered by exact cross-error, right-hand-side term count, expression-tree size, and canonical expression, after collapsing only exact integer no-ops.Fitted coefficient fields do not enter the equivalence test.
  • Table representation: Each representative table preserves the Agent-authored symbolic structure, marks equivalence to the generating law and final submission, and displays declared field dependencies.DSL integer literals N0 and N1 are typeset as 0 and 1.
  • Representative structures: One listed candidate has five independent fields with equation ut = C0(x, y)uxx + C1(x, y)uyy + C2(x, y)ux + C3(x, y)uy + C4(x, y)u and search error 0.451745.The candidate is ranked third in the displayed ordering.
  • Representative structures: Other listed alternatives include split directional diffusion and independent local fields, with search errors 0.459191 and 0.462034.These candidates appear among the structurally distinct low-error alternatives.

C Representative Harness Inputs

The representative harness instructs a scientific hypothesis Agent to infer one complete differential relation from two recorded trials using isolated numerical analysis and cross-trial validation.

  • Harness setup: The Agent’s goal is to infer and submit one complete governing differential relation explaining both recorded trials.The instruction frames the task without naming the PDE family or physical identity.
  • Harness setup: The harness directs the Agent to inspect problem, DSL, metadata, and raw arrays, then use numerical diagnostics and PDE commands to propose and score complete relations.The experimental record and measured columns are the available domain context.
  • Evaluation principles: Cross-trial error is treated as evidence rather than an oracle for symbolic similarity or uniqueness, with unsupported degrees of freedom penalized through scientific judgment.The instructions require comparison of coherent structural alternatives.
  • Submission: Before finishing, the Agent must run the history command and submit the judged-best candidate because a prose answer is not a valid submission.The evaluator does not complete a partial relation.

C.2 Representative PROBLEM.md

The task is to infer a complete differential relation from repeated measurements of one planar specimen under two preparation programs, despite an unknown response mechanism. The relation must transfer across both trials, while low numerical error alone does not establish a unique governing law.

  • A fixed rectangular probe array records one continuous scalar response at regular times, but the mechanism generating the readings is unknown.
  • Trial A and Trial B use the same specimen and probes after different preparation programs.
  • The objective is to determine a complete differential relation that transfers across both trials.
  • Low numerical error provides evidence but does not prove that the governing law is unique.

C.3 Candidate-language specification

The candidate language represents complete differential relations as recursive expression trees with explicitly declared coefficient fields, while the evaluation workflow preserves submitted structure and compares tested hypotheses using numerical evidence.

  • Candidate representation: A candidate is a complete differential relation written with the recorded column names in a specified text-file format.
  • Candidate representation: The recursive grammar supports fields, negation, grouping, binary operations, pointwise functions, and derivatives with respect to recorded coordinates or time.
  • Candidate representation: Unknown fields must enter linearly, with repeated names denoting the same field and distinct names denoting independently fitted fields.
  • Hypothesis evaluation: The evaluator performs no algebraic rewrite, simplification, expansion, factorization, or term completion, so operators remain at their submitted syntax-tree positions.
  • Search and selection: Search alternates independent construction with local refinement, requiring structurally distinct scored candidates before bounded final selection.
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