PDEFlow: Autonomous Agentic PDE Pipelines for Neural Operator Learning and Solver-Free Inference

PDEFlow is an agentic system designed to bridge the gap between natural-language problem descriptions and high-performance neural operator learning. By integrating stateful input handling with automated numerical simulation, the framework enables researchers to specify, simulate, and learn from differential equations without manual intervention at each step. The system is particularly suited for engineering design spaces where the same governing physics must be evaluated repeatedly under varying conditions.

The core of PDEFlow is a stateful input-handling agent that manages the evolution of a problem specification. Unlike traditional approaches that treat each user input as a fresh configuration, PDEFlow maintains a canonical JSON specification that is updated incrementally. A graph of specialized agents—including an orchestrator, extractor, validator, and repairer—ensures that every user edit is validated against the governing physics and schema requirements. This prevents silent errors and ensures that the final specification is always executable by the backend solver.

Once a specification is validated, the data-generation module uses the FEniCSx finite-element backend to solve the governing equations. This module handles the sampling of parameters, boundary conditions, and forcing terms, storing the results as operator-ready tensors. The training module then utilizes a registry-based interface to train neural operators, such as Bayesian DeepONet, on these datasets. This modular design allows researchers to swap different neural operator architectures without modifying the surrounding pipeline. Finally, the inference module enables fast, solver-free predictions from trained checkpoints, significantly accelerating design exploration and uncertainty quantification.

Scientific workflows often suffer from the high cost of manual setup and the fragility of repeated solver runs. PDEFlow addresses these bottlenecks by automating the translation of scientific intent into executable code and replacing expensive numerical solves with fast neural operator approximations. By preserving intermediate artifacts and maintaining a rigorous state, the framework provides a reliable, repeatable path for large-scale physics-based design and analysis.