Lang-PINN: From Language to Physics-Informed Neural Networks via a Multi-Agent Framework

TL;DR

Lang-PINN tackles the challenge of building trainable PINNs directly from natural-language task descriptions by introducing a four-agent system that translates NL into PDEs (PDE Agent), selects PDE-aware PINN architectures (PINN Agent), generates modular executable code (Code Agent), and iteratively diagnoses and fixes issues via runtime feedback (Feedback Agent). A key contribution is the Task2PDE dataset, enabling systematic evaluation of semantic-to-symbol grounding and end-to-end verification. Empirical results on the PINNacle suite show Lang-PINN achieving up to $10^{3}$–$10^{5}$-fold reductions in MSE, over a $50\%$ increase in end-to-end execution success, and up to $74\%$ faster convergence compared with strong baselines, underscoring the value of multi-agent coordination and verification in scientific AI pipelines. The work demonstrates a practical path to robust, verifiable PINN pipelines from plain-language prompts and suggests promising directions for multi-physics extensions and real-world noisy data scenarios.

Abstract

Physics-informed neural networks (PINNs) provide a powerful approach for solving partial differential equations (PDEs), but constructing a usable PINN remains labor-intensive and error-prone. Scientists must interpret problems as PDE formulations, design architectures and loss functions, and implement stable training pipelines. Existing large language model (LLM) based approaches address isolated steps such as code generation or architecture suggestion, but typically assume a formal PDE is already specified and therefore lack an end-to-end perspective. We present Lang-PINN, an LLM-driven multi-agent system that builds trainable PINNs directly from natural language task descriptions. Lang-PINN coordinates four complementary agents: a PDE Agent that parses task descriptions into symbolic PDEs, a PINN Agent that selects architectures, a Code Agent that generates modular implementations, and a Feedback Agent that executes and diagnoses errors for iterative refinement. This design transforms informal task statements into executable and verifiable PINN code. Experiments show that Lang-PINN achieves substantially lower errors and greater robustness than competitive baselines: mean squared error (MSE) is reduced by up to 3--5 orders of magnitude, end-to-end execution success improves by more than 50\%, and reduces time overhead by up to 74\%.

Figures (9)

• Figure 1: System overview of Lang-PINN. The framework decomposes end-to-end PINN design into four agents: PDE Agent (canonical PDE formulation), PINN Agent (training-free architecture selection), Code Agent (modularized code generation), and Feedback Agent (runtime error analysis and multi-dimensional evaluation). Iterative refinement with feedback forms a closed loop, yielding reliable and executable PINN programs from natural language descriptions.
• Figure 2: Impact of linguistic complexity on PDE translation. Accuracy is reported across four levels of description difficulty using Symbolic Equivalence and Semantic Consistency scores.
• Figure 3: Comparative MSE of different PINN architectures on representative PDEs. Results are shown in log scale for clarity.
• Figure 4: Comparative Success Rate(%) of different code generation paradigms (monolithic vs. modular)on six PDEs.
• Figure 5: Comparative success rates (%)across 1D/2D/3D/ND.