# lululxvi/deepxde

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3,874 stars · 919 forks · Python · lgpl-2.1

## Links

- GitHub: https://github.com/lululxvi/deepxde
- Homepage: https://deepxde.readthedocs.io
- awesome-repositories: https://awesome-repositories.com/repository/lululxvi-deepxde.md

## Topics

`deep-learning` `deeponet` `jax` `multi-fidelity-data` `neural-network` `operator` `paddle` `pde` `physics-informed-learning` `pinn` `pytorch` `scientific-machine-learning` `tensorflow`

## Description

DeepXDE is a scientific machine learning library and deep learning PDE solver used to compute solutions for forward and inverse ordinary, partial, and integro-differential equations. It functions as a physics-informed neural network library that embeds physical laws and boundary conditions directly into the neural network loss function.

The project provides a deep operator network framework for learning operator mappings that approximate relationships between functions in multiphysics problems. It is implemented as a multi-backend tensor library, allowing the system to switch between different tensor backends for hardware-accelerated numerical operations.

The library's capabilities cover automatic differentiation, boundary condition enforcement, and the definition of complex geometries using constructive solid geometry. It also includes tools for multi-fidelity data modeling, prediction uncertainty quantification, and distributed model training across multiple GPUs.

## Tags

### Artificial Intelligence & ML

- [Physics-Informed Architectures](https://awesome-repositories.com/f/artificial-intelligence-ml/neural-network-architectures/physics-informed-architectures.md) — Implements a physics-informed neural network library for solving forward and inverse differential equations.
- [Automatic Differentiation](https://awesome-repositories.com/f/artificial-intelligence-ml/automatic-differentiation.md) — Provides an automatic differentiation engine supporting both forward and reverse mode for evaluating differential operators.
- [Automatic Differentiation Frameworks](https://awesome-repositories.com/f/artificial-intelligence-ml/automatic-differentiation-frameworks.md) — Provides a comprehensive system for computing derivatives using forward and reverse mode differentiation.
- [Deep Operator Network Frameworks](https://awesome-repositories.com/f/artificial-intelligence-ml/deep-operator-network-frameworks.md) — Provides a dedicated framework for learning operator mappings that approximate function relationships in multiphysics.
- [Deep Operator Networks](https://awesome-repositories.com/f/artificial-intelligence-ml/deep-operator-networks.md) — Implements deep operator networks to approximate relationships between functions in multiphysics problems. ([source](https://deepxde.readthedocs.io))
- [Model Performance Evaluators](https://awesome-repositories.com/f/artificial-intelligence-ml/model-performance-evaluators.md) — Measures prediction accuracy using relative error, absolute percentage error, and other statistical metrics. ([source](https://deepxde.readthedocs.io/en/latest/modules/deepxde.html))
- [Model Predictions](https://awesome-repositories.com/f/artificial-intelligence-ml/model-predictions.md) — Generates predictions for input samples and computes residuals of differential operators for validation. ([source](https://deepxde.readthedocs.io/en/latest/modules/deepxde.html))
- [Operator Mapping Learning](https://awesome-repositories.com/f/artificial-intelligence-ml/operator-mapping-learning.md) — Trains neural networks to approximate mathematical operators that map functions to other functions using aligned datasets. ([source](https://deepxde.readthedocs.io/en/latest/demos/operator.html))
- [Loss Function Calculators](https://awesome-repositories.com/f/artificial-intelligence-ml/prediction-visualization/loss-function-calculators.md) — Computes the difference between predicted and true values using loss functions like mean squared error. ([source](https://deepxde.readthedocs.io/en/latest/modules/deepxde.html))
- [Tensor Computation Backends](https://awesome-repositories.com/f/artificial-intelligence-ml/tensor-computation-backends.md) — Implements a backend-agnostic wrapper that allows switching between TensorFlow, PyTorch, and JAX tensor backends.
- [Tensor Libraries](https://awesome-repositories.com/f/artificial-intelligence-ml/tensor-libraries.md) — Provides a multi-backend tensor library for hardware-accelerated scientific computing and neural network operations.
- [Distributed Training](https://awesome-repositories.com/f/artificial-intelligence-ml/distributed-training.md) — Splits training workloads across multiple GPUs using data-parallel processing to accelerate computation. ([source](https://deepxde.readthedocs.io/en/latest/))
- [Residual-Based Adaptive Resampling](https://awesome-repositories.com/f/artificial-intelligence-ml/machine-learning/infrastructure/machine-learning-resampling/residual-based-adaptive-resampling.md) — Implements adaptive resampling of training coordinates to improve accuracy in high-residual regions of the spatial domain.
- [Multi-Fidelity Optimization](https://awesome-repositories.com/f/artificial-intelligence-ml/multi-fidelity-optimization.md) — Integrates multi-fidelity data to improve the overall accuracy of the learned mathematical models. ([source](https://deepxde.readthedocs.io/en/latest/demos/function.html))
- [Multi-Fidelity Data Modeling](https://awesome-repositories.com/f/artificial-intelligence-ml/multi-fidelity-optimization/multi-fidelity-data-modeling.md) — Improves model accuracy by combining low- and high-fidelity datasets through transfer learning.
- [Multi-Fidelity Training](https://awesome-repositories.com/f/artificial-intelligence-ml/multi-fidelity-optimization/multi-fidelity-training.md) — Combines data of varying accuracy levels through transfer learning across scales to enhance model performance. ([source](https://deepxde.readthedocs.io/en/latest/user/faq.html))
- [Custom Architectures](https://awesome-repositories.com/f/artificial-intelligence-ml/neural-networks/custom-architectures.md) — Implements specialized neural network structures and custom loss functions tailored for complex mathematical problems. ([source](https://deepxde.readthedocs.io/en/latest/user/faq.html))
- [Multi-Fidelity](https://awesome-repositories.com/f/artificial-intelligence-ml/transfer-learning/multi-fidelity.md) — Combines low- and high-accuracy datasets through transfer learning to improve the precision of physical system models.

### Scientific & Mathematical Computing

- [Differential Equation Solvers](https://awesome-repositories.com/f/scientific-mathematical-computing/differential-equation-solvers.md) — Computes numerical solutions for forward and inverse ordinary, partial, and integro differential equations.
- [Boundary Condition Enforcement](https://awesome-repositories.com/f/scientific-mathematical-computing/boundary-condition-enforcement.md) — Implements constraints like Dirichlet or Neumann using hard constraints or approximate distance functions. ([source](https://cdn.jsdelivr.net/gh/lululxvi/deepxde@master/README.md))
- [Computational Backend Integrations](https://awesome-repositories.com/f/scientific-mathematical-computing/computational-backend-integrations.md) — Allows the selection of underlying tensor libraries for numerical operations via environment variables or detection. ([source](https://deepxde.readthedocs.io/en/latest/user/installation.html))
- [Deep Operator Networks](https://awesome-repositories.com/f/scientific-mathematical-computing/deep-operator-networks.md) — Implements deep operator networks to approximate mappings between functions in multiphysics problems.
- [Forward PDE Solvers](https://awesome-repositories.com/f/scientific-mathematical-computing/differential-equation-solvers/forward-pde-solvers.md) — Computes solutions for differential equations given specified parameters and boundary conditions. ([source](https://deepxde.readthedocs.io/_sources/index.rst.txt))
- [Inverse PDE Solvers](https://awesome-repositories.com/f/scientific-mathematical-computing/differential-equation-solvers/inverse-pde-solvers.md) — Estimates unknown parameters or forcing fields within differential equations by training on observed data. ([source](https://deepxde.readthedocs.io/_sources/index.rst.txt))
- [Physics-Informed Operator Solving](https://awesome-repositories.com/f/scientific-mathematical-computing/differential-equation-solvers/physics-informed-operator-solving.md) — Combines operator learning with physics-based constraints to solve differential equations across spatial dimensions. ([source](https://deepxde.readthedocs.io/en/latest/demos/operator.html))
- [Time-Independent PDE Solvers](https://awesome-repositories.com/f/scientific-mathematical-computing/differential-equation-solvers/time-independent-pde-solvers.md) — Computes steady-state solutions for partial differential equations across various boundary conditions. ([source](https://deepxde.readthedocs.io/en/latest/demos/pinn_forward.html))
- [Automatic Differentiation](https://awesome-repositories.com/f/scientific-mathematical-computing/high-performance-execution-environments/scientific-computing-platforms/scientific-computing/automatic-differentiation.md) — Calculates derivatives using reverse-mode backpropagation, forward-mode, or zero coordinate shift methods. ([source](https://deepxde.readthedocs.io))
- [Scientific Machine Learning](https://awesome-repositories.com/f/scientific-mathematical-computing/scientific-machine-learning.md) — Combines data-driven neural networks with mathematical constraints to model complex physical systems.
- [Fractional PDE Solving](https://awesome-repositories.com/f/scientific-mathematical-computing/differential-equation-solvers/fractional-pde-solving.md) — Computes solutions for partial differential equations involving fractional order derivatives across multiple dimensions. ([source](https://deepxde.readthedocs.io/en/latest/demos/pinn_forward.html))
- [Integro-Differential Equation Solving](https://awesome-repositories.com/f/scientific-mathematical-computing/differential-equation-solvers/integro-differential-equation-solving.md) — Calculates solutions for equations containing both derivatives and integrals, such as Volterra types. ([source](https://deepxde.readthedocs.io/en/latest/demos/pinn_forward.html))
- [Ordinary Differential Equation Solving](https://awesome-repositories.com/f/scientific-mathematical-computing/differential-equation-solvers/ordinary-differential-equation-solving.md) — Computes solutions for systems of ordinary differential equations including second order systems. ([source](https://deepxde.readthedocs.io/en/latest/demos/pinn_forward.html))
- [Mathematical Function Approximations](https://awesome-repositories.com/f/scientific-mathematical-computing/mathematical-function-approximations.md) — Learns mathematical mappings using explicit formulas or datasets to represent target functions. ([source](https://deepxde.readthedocs.io/en/latest/demos/function.html))
- [Mathematical Point Sampling](https://awesome-repositories.com/f/scientific-mathematical-computing/mathematical-point-sampling.md) — Generates training datasets using specialized mathematical sampling sequences like Latin Hypercube and Sobol. ([source](https://cdn.jsdelivr.net/gh/lululxvi/deepxde@master/README.md))
- [Constructive Solid Geometry Operations](https://awesome-repositories.com/f/scientific-mathematical-computing/numerical-mathematical-foundations/computational-geometry/geometric-operations/constructive-solid-geometry-operations.md) — Uses constructive solid geometry and boolean operations to define complex spatial domains for differential equations.

### Part of an Awesome List

- [Neural Networks and Deep Learning](https://awesome-repositories.com/f/awesome-lists/ai/neural-networks-and-deep-learning.md) — Implements deep operator learning to approximate operator mappings for complex multiphysics problems.
- [Physics and PDE Solvers](https://awesome-repositories.com/f/awesome-lists/ai/physics-and-pde-solvers.md) — Functions as a deep learning PDE solver for partial, ordinary, and integro differential equations.
- [Time-Dependent PDE Solvers](https://awesome-repositories.com/f/awesome-lists/ai/physics-and-pde-solvers/time-dependent-pde-solvers.md) — Implements solvers for evolving physical systems using adaptive refinement and training point resampling. ([source](https://deepxde.readthedocs.io/en/latest/demos/pinn_forward.html))