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

2026-07-06

AI Summary

This work proposes the first end-to-end autonomous agent framework that automatically translates user-specified ordinary and partial differential equation (ODE/PDE) problems—expressed in natural language—into trainable and inferable neural operators. The approach employs a stateful input graph to parse semantic intent, invokes the FEniCSx finite element solver to generate high-fidelity training data, and trains a multi-branch Bayesian DeepONet architecture to learn the underlying solution operator. The system supports interactive multi-turn editing, parameterized data generation, and modular deployment, substantially enhancing automation and reproducibility in scientific machine learning workflows. Evaluated across multiple ODE/PDE benchmarks, the framework achieves efficient, solver-free predictions without reliance on traditional numerical solvers.

Abstract

We present PDEFlow, an autonomous agentic framework that turns user-level ODE and PDE descriptions into solver-backed neural-operator pipelines. The workflow links problem specification, data generation, operator training, and checkpoint-based inference. A stateful input graph converts multi-turn natural-language input and user edits into validated problem specifications. The data-generation module then samples parameters, solves the configured governing-equation with FEniCSx finite-element backend, and stores the solutions as operator-ready tensors. The training and inference stages use a registry-based interface, allowing different neural operators to be trained and deployed without changing the surrounding pipeline. In the current implementation, we instantiate this interface with a multi-branch Bayesian DeepONet. Experiments on benchmark ODE and PDE tasks show that PDEFlow can construct valid specifications, generate solver-backed datasets, train neural operators across steady and transient problem classes, and provide solver-free predictions from saved checkpoints. The framework is designed for repeatable scientific and engineering workflows where many related physics configurations must be specified, simulated, learned, and queried with minimal manual intervention.

Problem

• neural operator
• PDE
• solver-free inference
• autonomous framework
• scientific workflow

Innovation

• autonomous agentic framework
• neural operator learning
• solver-free inference
• Bayesian DeepONet
• FEniCSx