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

cs.AIAug 19, 2026

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

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

Organizations: Fujian University of Technology · Fujian University of Technology Fujian Provincial Key Laboratory of Big Data Mining and Applications · Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences

Abstract

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.

Figures & tables

Figure 1: Positioning of equation-discovery methods by structure and coefficient regime. Labels mark 25 representative works; faint markers show the remaining 62 publications. Color indicates publication year. Within-cell placement has no quantitative meaning. HER-PDE targets open-form PDE discovery with nonparametric, time-invariant spatial coefficient fields.

Command	Agent-visible operation and response	Cost
pde status	Reports the unique-evaluation budget, submission state, and current archive winner.	None
pde validate <file>	Parses a complete candidate and checks the candidate-language and evolution-law constraints without fitting fields.	None
pde score <file>	Canonicalizes and evaluates one complete candidate. The response contains its identifier, validity or stable failure code, directional transfer errors, symmetric score, rank, cache state, and counters.	1/new
pde score-batch <files>	Applies score independently to up to 32 candidates; batching changes throughput but not candidate semantics.	1/new
pde history	Returns the scored archive in chronological or ranked order without creating a new evaluation.	None
pde evidence <id>	Exports the candidate’s already-fitted coefficient fields on common anchors, together with conditioning, selected hyperparameters, and cross-trajectory stability diagnostics.	None

Table 1: Agent-visible HEI interaction protocol. A unique evaluation is charged only when HEI scores a previously unseen canonical candidate.

Figure 2: Overview of the scientific-agent workflow. (a) The workspace contains two noisy trajectories, experimental context, the candidate language, and persistent research files. (b) The Agent drives numerical tools and HEI through a common terminal interface. (c) During Phase I, it analyzes the observations and constructs an initial archive of at least three distinct HEI-scored hypotheses. (d) During Phase II, global hypothesis exploration and local hypothesis refinement generate complete candidates; HEI returns structured evaluation feedback until the Agent submits a final hypothesis or exhausts the search budget.

Case	Generating law	Agent-selected law	Structural relation	Search	Sealed
Groundwater flow	$h_{t}=\nabla\!\cdot\!\left((K/S_{s})\nabla h\right)$	$\begin{aligned} h_{t}={}&\partial_{x}(C_{0}h_{x})\\ &+\partial_{y}(C_{0}h_{y})\end{aligned}$	Exact; $C_{0}=K/S_{s}$	0.5140	0.1226
Variable diffusion	$u_{t}=a\Delta u$	$u_{t}=C_{0}(u_{xx}+u_{yy})$	Exact; $C_{0}=a$	0.2093	0.2890
Reaction–diffusion	$u_{t}=\nabla\!\cdot(D\nabla u)-ku$	$\begin{aligned} u_{t}={}&C_{0,x}u_{x}+C_{0}u_{xx}\\ &+C_{0,y}u_{y}+C_{0}u_{yy}\\ &+C_{1}u\end{aligned}$	Product-rule equivalent; $C_{0}=D$ , $C_{1}=-k$	0.4506	0.4272
Advection–diffusion	$u_{t}=D\Delta u-\bm{v}\!\cdot\nabla u$	$\begin{aligned} u_{t}={}&C_{0}(u_{xx}+u_{yy})\\ &+C_{1}u_{x}+C_{2}u_{y}\end{aligned}$	Signed-field equivalent; $C_{0}=D$ , $(C_{1},C_{2})=-\bm{v}$	0.4944	0.5114
Orthotropic diffusion	$\begin{aligned} u_{t}={}&\partial_{x}(D_{x}u_{x})\\ &+\partial_{y}(D_{y}u_{y})\end{aligned}$	$\begin{aligned} u_{t}={}&\partial_{x}(C_{0}u_{x})\\ &+\partial_{y}(C_{1}u_{y})\end{aligned}$	Exact; $(C_{0},C_{1})=(D_{x},D_{y})$	0.2041	0.2761

Table 2: Five registered Agent discoveries. Search reports the HEI score on the third temporal interval; sealed reports the post-submission score on the final interval. The Agent-selected equations are the submitted structures rendered in mathematical notation.

Figure 3: Agent search trajectories for the five positive discovery cases. Gray points are valid candidates, the blue line is the best-so-far HEI error, red crosses are invalid evaluations without an imputed score, and the gold star is the first generating-structure candidate ultimately submitted by the Agent. For Groundwater, the hollow diamond is the lower-search-error archive winner with an added unit

y

-gradient term (its

x

-gradient multiplier is zero); the Agent instead selects the generating conservative operator. Target labels are assigned retrospectively and are not visible during search.

Figure 4: Noisy observations and recovered coefficient fields. (a) The central recorded time slice from trajectory A for each case; all state observations contain $5\%$ relative noise. (b) Ground truth and the HEI fit induced by the Agent-submitted law for all nine coefficient fields. Truth and fit are evaluated at identical anchors and displayed without spatial smoothing; every pair shares its color scale. Here $r$ is Pearson correlation and $E_{\mathrm{rel}}$ is defined by Eq. ( 31 ).

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix


Stage	Candidate or test	$E_{\mathrm{search}}$	Agent update
Phase I	Exploratory derivatives and cross-trial statistics	—	Enumerated diffusion, wave, advection, and reaction–diffusion as competing explanations.
Phase I	$u_{tt}=0\Delta u$	invalid	Used HEI’s language failure to rule out the second-order-time branch and restrict subsequent proposals to measured $u_{t}$ .
Eval. 1	$u_{t}=0\Delta u$	1.0246	The Agent-authored zero multiplier makes this the effective null law $u_{t}=0$ , establishing the first numerical baseline.
Eval. 2	$u_{t}=C_{0}\Delta u$	1.4641	Found that spatially varying diffusivity alone did not explain transfer between the two preparations.
Eval. 8	$u_{t}=C_{1}u_{x}+C_{2}u_{y}$	4.2552	Ruled out heterogeneous transport without a second-order component.

Appendix

Table 3: Detailed advection–diffusion decision trace. “Eval.” is the chronological index of a distinct HEI-scored candidate.


Case	Chronological structural progression	Selection evidence
Groundwater flow	Flat heterogeneous diffusion $C_{0}\Delta h$ $(2,1.0598)$ was followed by the shared conservative construction $\nabla\!\cdot(C_{0}\nabla h)$ $(11,0.5140)$ . Reaction, source, and heterogeneous transport extensions all increased the error.	A child with a unit $y$ -gradient term reached 0.5132, but added unsupported structure. The Agent submitted the compact parent; its sealed error, 0.122599, was also slightly below the child’s 0.122641.
Variable diffusion	The first candidate, $C_{0}\Delta u$ $(1,0.2093)$ , remained rank one while the Agent tested split directional fields (0.5045), reaction (0.3767), one-coordinate dependence (0.2628), and nonlinear modulation (0.2287).	The Agent retained the first candidate after 32 distinct scores. Its sealed error was 0.2890.
Reaction–diffusion	Reaction alone $(1,10.1620)$ gave way to split diffusion plus reaction $(3,0.5403)$ , then to a fully expanded local model $(5,0.4517)$ . Linking the derivative and second-order terms through one shared diffusivity produced the conservative product-rule expansion $(7,0.4506)$ .	Independent directional diffusivities scored 0.4592 and extra advection scored 0.4507. The Agent submitted the shared-field expansion; its sealed error was 0.4272.
Advection–diffusion	Diffusion-only and transport-only candidates exceeded 1.0. Their joint construction with reaction reached 0.6588 at evaluation 31; deleting the reaction produced the exact three-field structure at evaluation 37 with error 0.4944.	Missing-direction, linked-drift, nonlinear-reaction, conservative-placement, and split-diffusion controls were all worse. The sealed error was 0.5114.
Orthotropic diffusion	Shared isotropic diffusion $(2,0.3519)$ improved to a two-field nonconservative diagonal form $(4,0.2984)$ . Placing each field inside its directional derivative produced $\partial_{x}(C_{0}u_{x})+\partial_{y}(C_{1}u_{y})$ $(15,0.2041)$ .	The explicit shared-field product-rule expansion at evaluation 22 had the identical score. The Agent submitted the compact nested form; its sealed error was 0.2761.

Appendix

Table 4: Compressed decision traces for the five registered discoveries. Parenthesized pairs give the HEI evaluation index and search error.

Rank	Eval.	Candidate equation	$E_{\mathrm{search}}$	Structural audit
1	23	$h_{t}=\partial_{x}(C_{0}(x,y)h_{x})+\partial_{y}(C_{0}(x,y)h_{y})+0h_{x}+1h_{y}$	0.513184	Integer-weighted gradient extension
2	11	$\bm{h_{t}=\partial_{x}(C_{0}(x,y)h_{x})+\partial_{y}(C_{0}(x,y)h_{y})}$	0.513982	$\bm{\checkmark}$ Target; $\bm{\dagger}$ submitted
3	14	$h_{t}=\partial_{x}(C_{0}(x,y)h_{x})+\partial_{y}(C_{0}(x,y)h_{y})+C_{1}(x,y)h$	0.691988	Extra spatial reaction
4	15	$h_{t}=\partial_{x}(C_{0}(x,y)h_{x})+\partial_{y}(C_{0}(x,y)h_{y})+C_{1}(x,y)$	0.692116	Extra spatial source
5	1	$h_{t}=0\Delta h$	1.001298	Agent-authored null law

Appendix

Table 5: Groundwater flow: five lowest-error structurally distinct valid candidates.

Rank	Eval.	Candidate equation	$E_{\mathrm{search}}$	Structural audit
1	1	$\bm{u_{t}=C_{0}(x,y)\Delta u}$	0.209334	$\bm{\checkmark}$ Target; $\bm{\dagger}$ submitted
2	32	$u_{t}=C_{0}(x,y)\Delta u(1+u^{2})$	0.228653	Nonlinear modulation
3	19	$u_{t}=C_{0}(x)\Delta u$	0.262803	Missing $y$ dependence
4	21	$u_{t}=C_{0}(x,y)\Delta u+C_{1}(x,y)u_{xx}u_{yy}$	0.304122	Extra nonlinear term
5	17	$u_{t}=C_{0}(x,y)\Delta u+C_{1}(x,y)\tanh(u)$	0.365506	Extra nonlinear reaction

Appendix

Table 6: Variable diffusion: five lowest-error structurally distinct valid candidates.

Rank	Eval.	Candidate equation	$E_{\mathrm{search}}$	Structural audit
1	7	$\begin{aligned} u_{t}={}&\partial_{x}C_{0}(x,y)u_{x}+C_{0}(x,y)u_{xx}\\ &+\partial_{y}C_{0}(x,y)u_{y}+C_{0}(x,y)u_{yy}+C_{1}(x,y)u\end{aligned}$	0.450614	$\bm{\checkmark}$ Target; $\bm{\dagger}$ submitted
2	18	$\begin{aligned} u_{t}={}&\partial_{x}C_{0}(x,y)u_{x}+C_{0}(x,y)u_{xx}\\ &+\partial_{y}C_{0}(x,y)u_{y}+C_{0}(x,y)u_{yy}+C_{1}(x,y)u\\ &+C_{2}(x,y)u_{x}+C_{3}(x,y)u_{y}\end{aligned}$	0.450719	Extra spatial transport
3	5	$\begin{aligned} u_{t}={}&C_{0}(x,y)u_{xx}+C_{1}(x,y)u_{yy}+C_{2}(x,y)u_{x}\\ &+C_{3}(x,y)u_{y}+C_{4}(x,y)u\end{aligned}$	0.451745	Five independent fields
4	12	$\begin{aligned} u_{t}={}&\partial_{x}C_{0}(x,y)u_{x}+C_{0}(x,y)u_{xx}+\partial_{y}C_{1}(x,y)u_{y}\\ &+C_{1}(x,y)u_{yy}+C_{2}(x,y)u\end{aligned}$	0.459191	Split directional diffusion
5	16	$u_{t}=C_{0}(x,y)\Delta u+C_{1}(x,y)u_{x}+C_{2}(x,y)u_{y}+C_{3}(x,y)u$	0.462034	Independent local fields

Appendix

Table 7: Reaction–diffusion: five lowest-error structurally distinct valid candidates.

Rank	Eval.	Candidate equation	$E_{\mathrm{search}}$	Structural audit
1	37	$\bm{u_{t}=C_{0}(x,y)\Delta u+C_{1}(x,y)u_{x}+C_{2}(x,y)u_{y}}$	0.494444	$\bm{\checkmark}$ Target; $\bm{\dagger}$ submitted
2	62	$\begin{aligned} u_{t}={}&\partial_{x}(C_{0}(x,y)u_{x})+\partial_{y}(C_{0}(x,y)u_{y})\\ &+C_{1}(x,y)u_{x}+C_{2}(x,y)u_{y}\end{aligned}$	0.547559	Conservative diffusion placement
3	52	$u_{t}=C_{0}(x,y)\Delta u+C_{1}(x,y)u_{x}+C_{2}(x,y)u_{y}+C_{3}(x,y)\sin(u)$	0.625828	Extra nonlinear reaction
4	31	$u_{t}=C_{0}(x,y)\Delta u+C_{1}(x,y)u_{x}+C_{2}(x,y)u_{y}+C_{3}(x,y)u$	0.658754	Extra linear reaction
5	39	$\begin{aligned} u_{t}={}&C_{0}(x,y)u_{xx}+C_{1}(x,y)u_{yy}+C_{2}(x,y)u_{x}\\ &+C_{3}(x,y)u_{y}+C_{4}(x,y)u\end{aligned}$	0.675537	Split diffusion and reaction

Appendix

Table 8: Advection–diffusion: five lowest-error structurally distinct valid candidates.

Rank	Eval.	Candidate equation	$E_{\mathrm{search}}$	Structural audit
1	15	$\bm{u_{t}=\partial_{x}(C_{0}(x,y)u_{x})+\partial_{y}(C_{1}(x,y)u_{y})}$	0.204067	$\bm{\checkmark}$ Target; $\bm{\dagger}$ submitted
2	22	$\begin{aligned} u_{t}={}&C_{0}(x,y)u_{xx}+C_{1}(x,y)u_{yy}\\ &+\partial_{x}C_{0}(x,y)u_{x}+\partial_{y}C_{1}(x,y)u_{y}\end{aligned}$	0.204067	$\checkmark$ Product-rule equivalent
3	29	$\begin{aligned} u_{t}={}&\partial_{x}(C_{0}(x,y)u_{x})+\partial_{y}(C_{1}(x,y)u_{y})\\ &+C_{2}(x,y)u_{xx}u_{yy}\end{aligned}$	0.247207	Extra nonlinear term
4	4	$u_{t}=C_{0}(x,y)u_{xx}+C_{1}(x,y)u_{yy}$	0.298445	Missing field derivatives
5	2	$u_{t}=C_{0}(x,y)(u_{xx}+u_{yy})$	0.351889	Shared isotropic field

Appendix

Table 9: Orthotropic diffusion: five lowest-error structurally distinct valid candidates.


Work / citation	Structure	Coefficient	Note
Equations of motion ( Crutchfield and McNamara 1987 )	Mixed	Constant coefficients	Structure depends on representation.
GP structure ID ( Gray et al. 1998 )	Open-form library	Constant coefficients
GP-ODE ( Cao et al. 2000 )	Open-form library	Constant coefficients
ARE ( Bongard and Lipson 2007 )	Open-form library	Constant coefficients
Free-form laws ( Schmidt and Lipson 2009 )	Open-form library	Constant coefficients

Appendix

Table 10: Audited literature-positioning pool (87 publications).

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

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

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

PDE Discovery AI Agents for Scientific Discovery Large Language Model-Guided Scientific Discovery

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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.

Abstract

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