DeepXDE is a library for scientific machine learning that implements physics-informed neural networks and operator learning methods to solve differential equations and learn solution operators from data.
The library addresses the challenge of combining neural networks with physical constraints and domain knowledge. It solves forward and inverse problems involving ordinary differential equations, partial differential equations, integro-differential equations, fractional PDEs, and stochastic PDEs by encoding physical laws directly into the network training process. Beyond standard physics-informed neural networks, the tool implements specialized variants including residual-based adaptive sampling to focus training on regions of high error, gradient-enhanced networks that incorporate derivative information, and multi-scale Fourier features to improve accuracy on problems with multiple length scales. It also provides deep operator networks that learn mappings between function spaces rather than solving individual instances, enabling rapid evaluation across different inputs and parameters.
Adoption suits researchers and engineers working on inverse design, topology optimization, multiphysics problems, or scenarios where data is expensive or sparse. The tool is particularly valuable when physical equations are known but difficult to solve analytically or computationally. The library supports five tensor backends—TensorFlow 1.x, TensorFlow 2.x, PyTorch, JAX, and PaddlePaddle—allowing integration into existing workflows. Multifidelity learning capabilities enable combining high-accuracy but expensive simulations with cheaper approximations, and extensions like POD-DeepONet and MIONet handle multiple-input operators and reduced-order modeling.
The project maintains active development with regular implementation of recent algorithmic advances from the scientific computing literature. Contributions span both core algorithm implementations and backend support across multiple deep learning frameworks. The codebase includes comprehensive documentation with tutorials and examples demonstrating application to various problem classes.
