PINNForge: Execution-Grounded Evolutionary Design of Physics-Informed Neural Networks for PDE Solving via Large Language Models

Abstract

Physics-informed neural networks (PINNs) require coordinated choices over network representation, sampling, loss construction, and optimization, while effective configurations often vary substantially across partial differential equations (PDEs). Existing automated PINN design methods can search candidate configurations, but information revealed during actual training is still used mainly for evaluation rather than to improve subsequent design, leading to repeated trial-and-error and inefficient use of training budget. We propose PINNsForge, an LLM-driven evolutionary framework for execution-feedback-based automated PINN design. PINNsForge generates diverse candidate configurations from PDE-related prior knowledge, evaluates them through actual training, and feeds high-performing designs together with accumulated execution evidence back to the LLM. Guided by observed optimization behavior, the LLM then refines, recombines, and explores coupled PINN design components, forming a continual cycle of generation, execution, feedback, and evolution. Unlike one-shot search or evaluation-only feedback, PINNsForge progressively converts training experience into improved design decisions for the target PDE. Across 25 PDE benchmarks, PINNsForge achieves the lowest mean MSE on 24 tasks compared with RoPINN, PINNsFormer, and PINNsAgent. Ablation studies further confirm the importance of the PDE knowledge base, execution feedback, and evolutionary search: removing these components increases the mean MSE to 3.74$\times$, 12.10$\times$, and 10.10$\times$ that of the full PINNsForge, respectively.