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

Scientific and engineering studies often require repeated ODE or PDE simulations across changing parameters, initial conditions, or boundary conditions. Existing workflows typically separate problem specification, numerical simulation, dataset generation, neural-operator training, and inference, making repeated studies computationally expensive and difficult to maintain as the problem specification evolves. We present PDEFlow, an autonomous agentic framework that connects these stages in an end-to-end, solver-backed neural-operator pipeline. A stateful input graph converts multi-turn natural-language descriptions and user edits into validated problem specifications, while preserving changes through validated JSON patches. The data-generation module samples physical configurations from prescribed distributions, solves the governing equations using a FEniCSx finite-element backend, and stores operator-ready datasets. A registry-based interface supports neural-operator training and checkpoint-based inference; the current implementation uses a multi-branch Bayesian DeepONet for uncertainty-aware prediction. Across 70 scripted multi-turn specification scenarios, the complete system achieves 83.81 ± 2.97% specification-match accuracy over three stochastic evaluation passes. Experiments on 11 parameterised ODE and PDE benchmarks demonstrate solver-backed operator learning across steady and transient problems, with model-forward inference speedups of up to 31.85×. PDEFlow provides a reproducible workflow for rapidly specifying, simulating, learning, and querying families of related physical configurations with reduced manual intervention.