OpInf-LLM: Parametric PDE Solving with LLMs via Operator Inference

Zhuoyuan Wang; Hanjiang Hu; Xiyu Deng; Saviz Mowlavi; Yorie Nakahira

cs.LGFeb 2, 2026

OpInf-LLM: Parametric PDE Solving with LLMs via Operator Inference

Authors: Zhuoyuan Wang, Hanjiang Hu, Xiyu Deng, Saviz Mowlavi, Yorie Nakahira

Organizations: Carnegie Mellon University · Mitsubishi Electric Research Laboratories

Abstract

Solving diverse partial differential equations (PDEs) is fundamental in science and engineering. Large language models (LLMs) have demonstrated strong capabilities in code generation, symbolic reasoning, and tool use, but reliably solving PDEs across heterogeneous settings remains challenging. Prior work on LLM-based code generation and transformer-based foundation models for PDE learning has shown promising advances. However, a persistent trade-off between execution success rate and numerical accuracy arises, particularly when generalization to unseen parameters and boundary conditions is required. In this work, we propose OpInf-LLM, an LLM parametric PDE solving framework via operator inference. The proposed framework leverages small amounts of solution data to enable accurate prediction of diverse PDE instances, including unseen parameters and configurations, and provides seamless integration with LLMs for natural language task specification and physics-based reasoning of proper feature parameterization. Its low computational demands and unified solution pipeline further enable a high execution success rate across heterogeneous settings, opening new possibilities for generalizable reduced-order modeling in LLM-based PDE solving.

Figures & tables

Figure 1: Trade-offs in prediction error and solver execution success rate. The mean and variance are computed from the experimental results in Section 4 across the three benchmark problems considered in this study.

Figure 2: Overall diagram of the OpInf-LLM framework. In the offline stage, shared basis and reduced-order operators are obtained from limited amounts of PDE solution data via proper orthogonal decomposition (POD) and least-squares optimization. In the online stage, LLM parses natural language instructions, generates code or calls agentic tools for operator regression and reduced-order ODE integration, to predict PDE solutions of diverse parametric and boundary conditions, including configurations beyond the available dataset.

Method	Data	Train time (s)	Success rate $\uparrow$	Heat $\downarrow$	Burgers $\downarrow$	Cavity $\downarrow$
CodePDE (GPT-4.1)	–	–	49.9%	$\underline{1.59\mathrm{e}{-2}}$	$\bm{1.50\mathrm{e}{-1}}$	$1.41\mathrm{e}{0}$
CodePDE (GPT-4o)	–	–	39.9%	$1.60\mathrm{e}{-1}$	$1.23\mathrm{e}{0}$	$2.89\mathrm{e}{0}$
CodePDE (Gemini-2.0-flash)	–	–	47.6%	$1.74\mathrm{e}{-2}$	$9.56\mathrm{e}{-1}$	$1.55\mathrm{e}{0}$
MOL-LLM	449	14306	100.0%	$1.69\mathrm{e}{0}$	$1.44\mathrm{e}{0}$	$7.24\mathrm{e}{-1}$
MOL-LLM (large dataset)	15000	35915	100.0%	$4.87\mathrm{e}{0}$	$1.58\mathrm{e}{0}$	${6.44\mathrm{e}{-1}}$
LLM (GPT-4.1)	–	–	100.0%	$7.79\mathrm{e}{-1}$	$9.82\mathrm{e}{-1}$	$1.00\mathrm{e}{0}$

Table 1: Results summary. Average relative

L_{2}

error is reported for each equation. Bold denotes the best result, and underline denotes the second best.

\uparrow

higher is better,

\downarrow

lower is better.

Figure 5: Lid-driven cavity predictions for $\mathrm{Re}=80$ and $\mathrm{Re}=120$ at $t=1$ and $t=2$ . All plots share the same color scale. Results are shown for CodePDE, LLM and OpInf-LLM with GPT-4.1.

LLM model	Equation	Error $[0,T]$	Error $[T,2T]$
GPT-4o	Heat	$1.29\mathrm{e}{-2}$	$3.60\mathrm{e}{-2}$
	Burgers	$4.91\mathrm{e}{-1}$	$6.36\mathrm{e}{-1}$
	Cavity	$4.63\mathrm{e}{-2}$	$4.86\mathrm{e}{-2}$
Gemini-2.0 flash	Heat	$1.29\mathrm{e}{-2}$	$3.60\mathrm{e}{-2}$
	Burgers	$4.91\mathrm{e}{-1}$	$6.36\mathrm{e}{-1}$
	Cavity	$4.63\mathrm{e}{-2}$	$4.86\mathrm{e}{-2}$

Table 2: Prediction error of OpInf-LLM with extrapolation in time.

Method	Test $\mathrm{Re}$	Distribution	Feature	Error
OpInf	$110$	In-distribution	$\mathrm{Re}$	$3.57\mathrm{e}{-2}$
OpInf-LLM	$110$	In-distribution	$1/\mathrm{Re}$	$2.57\mathrm{e}{-2}$
OpInf	$200$	OOD	$\mathrm{Re}$	instability
OpInf-LLM	$200$	OOD	$1/\mathrm{Re}$	$7.47\mathrm{e}{-2}$

Table 3: Effect of feature parameterization on generalization for the lid-driven cavity problem.

Method	Exact Match	Field Accuracy
Static Parser	52% (26/50)	75.7% (109/144)
OpInf-LLM	100% (50/50)	100% (144/144)

Table 4: Natural language parsing performance comparison.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

\mathbf{Y}=\left[y\left(\bm{x},t_{1},\xi_{1}\right),\ldots,y\left(\bm{x},t_{K},\xi_{1}\right),y\left(\bm{x},t_{1},\xi_{2}\right),\ldots,y\left(\bm{x},t_{K},\xi_{2}\right),\ldots\right]

Appendix

Algorithm 1 OpInf-LLM

		Heat		Burgers		Cavity
Model	Success Rate	Attempt	Error	Attempt	Error	Attempt	Error
Gemini-2.0	99.2%	1	$1.29\mathrm{e}{-2}$	1	$4.91\mathrm{e}{-1}$	2	$3.88\mathrm{e}{-1}$
GPT-4o	99.2%	1	$1.87\mathrm{e}{-2}$	4	$4.91\mathrm{e}{-1}$	5	$3.26\mathrm{e}{-2}$
GPT-4.1	99.2%	1	$1.29\mathrm{e}{-2}$	1	$4.91\mathrm{e}{-1}$	1	$3.40\mathrm{e}{-2}$

Appendix

Table 5: Performance of OpInf-LLM under direct code generation without external tools.

Equation	POD	Energy (%)	Error $[0,T]$	Error $[T,2T]$
Burgers	4	88.01	$3.93\mathrm{e}{-1}$	$4.99\mathrm{e}{-1}$
Burgers	5	92.26	$3.44\mathrm{e}{-1}$	$4.48\mathrm{e}{-1}$
Burgers	6	95.63	$4.13\mathrm{e}{-1}$	$9.63\mathrm{e}{-1}$
Burgers	7	97.26	$3.52\mathrm{e}{-1}$	$5.41\mathrm{e}{-1}$
Burgers	8	98.31	$3.64\mathrm{e}{-1}$	$4.90\mathrm{e}{-1}$
Burgers	9	98.85	$4.70\mathrm{e}{-1}$	$6.46\mathrm{e}{-1}$

Appendix

Table 6: POD basis ablation for OpInf-LLM.

Equation	$\lambda$	Norm	Error $[0,T]$	Error $[T,2T]$
Burgers	$0$	1730.53	$2.25\mathrm{e}{1}$	$2.92\mathrm{e}{1}$
Burgers	$10^{-6}$	1730.53	$2.25\mathrm{e}{1}$	$2.92\mathrm{e}{1}$
Burgers	$10^{-5}$	1645.36	$2.25\mathrm{e}{1}$	$2.92\mathrm{e}{1}$
Burgers	$10^{-4}$	1645.28	$2.25\mathrm{e}{1}$	$2.92\mathrm{e}{1}$
Burgers	$10^{-3}$	1637.70	$2.26\mathrm{e}{1}$	$2.94\mathrm{e}{1}$
Burgers	$10^{-2}$	1377.88	$2.96\mathrm{e}{1}$	$3.90\mathrm{e}{1}$

Appendix

Table 7: Regularization intensity (

\lambda

) ablation for OpInf-LLM.

Panel of Figure 6 — Figure 6: Prediction comparison for the feature reasoning example.

Test parameter $\xi$	Raw- $\xi$ ROM	OpInf-LLM ROM	Oracle ROM
$0.75$	$3.0119\times 10^{-2}$	$4.5243\times 10^{-4}$	$4.5243\times 10^{-4}$
$1.60$	$7.7841\times 10^{-2}$	$5.2922\times 10^{-4}$	$5.2922\times 10^{-4}$
$2.00$	$2.8715\times 10^{0}$	$5.8492\times 10^{-4}$	$5.8492\times 10^{-4}$

Appendix

Table 8: Relative trajectory errors for the parametric ROM feature-identification test. The raw-

\xi

ROM uses the naive feature map

\{1,\xi\}

, while the OpInf-LLM ROM uses the feature map identified from the neutral prompt. The OpInf-LLM ROM recovers the oracle feature map and substantially improves test accuracy.

Figure 7: Trajectory-level relative error distributions for the heat equations.

Figure 8: Trajectory-level relative error distributions for the Burgers’ equations.

Figure 9: Trajectory-level relative error distributions for the lid-driven cavity equations.

Method	Data	Train time (s)	Success rate $\uparrow$	Heat $\downarrow$	Burgers $\downarrow$	Cavity $\downarrow$
Parametric OpInf reference	449	30	99.2%	$1.29\mathrm{e}{-2}$	$4.91\mathrm{e}{-1}$	$4.63\mathrm{e}{-2}$
OpInf-LLM (tool, GPT-4.1)	449	30	99.2%	$1.29\mathrm{e}{-2}$	$4.91\mathrm{e}{-1}$	$4.63\mathrm{e}{-2}$
OpInf-LLM (tool, GPT-4o)	449	30	99.2%	$1.29\mathrm{e}{-2}$	$4.91\mathrm{e}{-1}$	$4.63\mathrm{e}{-2}$
OpInf-LLM (tool, Gemini-2.0-flash)	449	30	99.2%	$1.29\mathrm{e}{-2}$	$4.91\mathrm{e}{-1}$	$4.63\mathrm{e}{-2}$
OpInf-LLM (codegen, GPT-4.1)	449	30	99.2%	$1.29\mathrm{e}{-2}$	$4.91\mathrm{e}{-1}$	$3.40\mathrm{e}{-2}$
OpInf-LLM (codegen, GPT-4o)	449	30	99.2%	$1.87\mathrm{e}{-2}$	$4.91\mathrm{e}{-1}$	$3.26\mathrm{e}{-2}$

Appendix

Table 9: Comparison with manually constructed parametric OpInf reference.

LLM model	Equation	Method	Parameter	Error $[0,T]$	Error $[T,2T]$	Success rate (%)
GPT-4.1	heat	interpolation	0.5	$8.70\mathrm{e}{-3}$	$5.97\mathrm{e}{-3}$	100
GPT-4.1	heat	interpolation	1.0	$6.07\mathrm{e}{-3}$	$5.24\mathrm{e}{-3}$	100
GPT-4.1	heat	interpolation	3.0	$2.22\mathrm{e}{-2}$	$1.17\mathrm{e}{-1}$	100
GPT-4.1	heat	regression	0.5	$8.70\mathrm{e}{-3}$	$5.97\mathrm{e}{-3}$	100
GPT-4.1	heat	regression	1.0	$1.02\mathrm{e}{-2}$	$3.08\mathrm{e}{-2}$	100
GPT-4.1	heat	regression	3.0	$1.99\mathrm{e}{-2}$	$7.12\mathrm{e}{-2}$	100

Appendix

Table 10: OpInf-LLM full results.

Equation	POD	Method	Error $[0,T]$	Error $[T,2T]$	Error $[0,2T]$
Heat	4	interpolation	$3.35\mathrm{e}{-2}$	$5.81\mathrm{e}{-2}$	$4.78\mathrm{e}{-2}$
Heat	4	regression	$1.96\mathrm{e}{-2}$	$2.39\mathrm{e}{-2}$	$2.21\mathrm{e}{-2}$
Heat	5	interpolation	$1.77\mathrm{e}{-2}$	$3.72\mathrm{e}{-2}$	$2.98\mathrm{e}{-2}$
Heat	5	regression	$3.32\mathrm{e}{-2}$	$7.87\mathrm{e}{-2}$	$6.03\mathrm{e}{-2}$
Heat	6	interpolation	$1.23\mathrm{e}{-2}$	$4.27\mathrm{e}{-2}$	$3.16\mathrm{e}{-2}$
Heat	6	regression	$1.29\mathrm{e}{-2}$	$3.60\mathrm{e}{-2}$	$2.72\mathrm{e}{-2}$

Appendix

Table 11: POD modes ablation for OpInf-LLM.

Equation	$\lambda$	Method	Error $[0,T]$	Error $[T,2T]$	Error $[0,2T]$
Heat	$0$	interpolation	$1.23\mathrm{e}{-2}$	$4.27\mathrm{e}{-2}$	$3.16\mathrm{e}{-2}$
Heat	$0$	regression	$1.29\mathrm{e}{-2}$	$3.60\mathrm{e}{-2}$	$2.72\mathrm{e}{-2}$
Heat	$10^{-6}$	interpolation	$1.23\mathrm{e}{-2}$	$4.27\mathrm{e}{-2}$	$3.16\mathrm{e}{-2}$
Heat	$10^{-6}$	regression	$1.29\mathrm{e}{-2}$	$3.60\mathrm{e}{-2}$	$2.72\mathrm{e}{-2}$
Heat	$10^{-5}$	interpolation	$1.23\mathrm{e}{-2}$	$4.27\mathrm{e}{-2}$	$3.16\mathrm{e}{-2}$
Heat	$10^{-5}$	regression	$1.29\mathrm{e}{-2}$	$3.60\mathrm{e}{-2}$	$2.72\mathrm{e}{-2}$

Appendix

Table 12: Regularization intensity (

\lambda

) ablation for OpInf-LLM.

Figure 10: Heat equation predictions of different $\nu$ .

Figure 11: Burgers’ equation predictions of different $\nu$ .

Figure 12: Cavity results across Reynolds numbers.

Figure 13: Cavity results across Reynolds numbers.

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Numerical methods and deep learning have advanced the modelling and simulation of systems governed by partial differential equations (PDEs), while formulating equations, configuring solvers, and translating simulations into design and control decisions continue to demand substantial domain expertise and computational resources. Large language models (LLMs) offer opportunities to automate and coordinate these tasks by combining scientific knowledge, mathematical reasoning, code generation, and tool use. Here we review LLM-assisted PDE research across three stages, including discovery, solving, and optimization. We examine how LLMs interact with physical knowledge, numerical methods, and computational feedback to formulate models, implement solution procedures, and guide design and control. We also review existing benchmarks and evaluation practices for assessing scientific validity, computational performance, and workflow reliability. Finally, we discuss three interconnected challenges and opportunities: developing mathematical and physical reasoning tailored to specific PDE problems; building transferable domain expertise through domain-specific and multimodal post-training; and integrating active experimental design with scientific validation for open-ended problems. These directions outline a path towards a more substantive role for LLMs in developing and testing scientific ideas and computational methods across PDE research.

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