index.rst

DeepXDE

DeepXDE is a library for scientific machine learning and physics-informed learning. DeepXDE includes the following algorithms:

• physics-informed neural network (PINN)

  • solving different problems

    • solving forward/inverse ordinary/partial differential equations (ODEs/PDEs) [SIAM Rev.]
    • solving forward/inverse integro-differential equations (IDEs) [SIAM Rev.]
    • fPINN: solving forward/inverse fractional PDEs (fPDEs) [SIAM J. Sci. Comput.]
    • NN-arbitrary polynomial chaos (NN-aPC): solving forward/inverse stochastic PDEs (sPDEs) [J. Comput. Phys.]
    • PINN with hard constraints (hPINN): solving inverse design/topology optimization [SIAM J. Sci. Comput.]

  • improving PINN accuracy

    • residual-based adaptive sampling [SIAM Rev., Comput. Methods Appl. Mech. Eng.]
    • gradient-enhanced PINN (gPINN) [Comput. Methods Appl. Mech. Eng.]
    • PINN with multi-scale Fourier features [Comput. Methods Appl. Mech. Eng.]

  • Slides, Video, Video in Chinese

• (physics-informed) deep operator network (DeepONet)

  • DeepONet: learning operators [Nat. Mach. Intell.]
  • DeepONet extensions, e.g., POD-DeepONet [Comput. Methods Appl. Mech. Eng.]
  • MIONet: learning multiple-input operators [SIAM J. Sci. Comput.]
  • Fourier-DeepONet [Comput. Methods Appl. Mech. Eng.], Fourier-MIONet [arXiv]
  • physics-informed DeepONet [Sci. Adv.]
  • multifidelity DeepONet [Phys. Rev. Research]
  • DeepM&Mnet: solving multiphysics and multiscale problems [J. Comput. Phys., J. Comput. Phys.]
  • Reliable extrapolation [Comput. Methods Appl. Mech. Eng.]

• multifidelity neural network (MFNN)

  • learning from multifidelity data [J. Comput. Phys., PNAS]

DeepXDE supports five tensor libraries as backends: TensorFlow 1.x (tensorflow.compat.v1 in TensorFlow 2.x), TensorFlow 2.x, PyTorch, JAX, and PaddlePaddle. For how to select one, see Working with different backends.

Documentation: ReadTheDocs

Features

DeepXDE has implemented many algorithms as shown above and supports many features:

• enables the user code to be compact, resembling closely the mathematical formulation.
• complex domain geometries without tyranny mesh generation. The primitive geometries are interval, triangle, rectangle, polygon, disk, ellipse, star-shaped, cuboid, sphere, hypercube, and hypersphere. Other geometries can be constructed as constructive solid geometry (CSG) using three boolean operations: union, difference, and intersection. DeepXDE also supports a geometry represented by a point cloud.
• 5 types of boundary conditions (BCs): Dirichlet, Neumann, Robin, periodic, and a general BC, which can be defined on an arbitrary domain or on a point set; and approximate distance functions for hard constraints.
• different neural networks: fully connected neural network (FNN), stacked FNN, residual neural network, (spatio-temporal) multi-scale Fourier feature networks, etc.
• many sampling methods: uniform, pseudorandom, Latin hypercube sampling, Halton sequence, Hammersley sequence, and Sobol sequence. The training points can keep the same during training or be resampled (adaptively) every certain iterations.
• 4 function spaces: power series, Chebyshev polynomial, Gaussian random field (1D/2D).
• data-parallel training on multiple GPUs.
• different optimizers: Adam, L-BFGS, etc.
• conveniently save the model during training, and load a trained model.
• callbacks to monitor the internal states and statistics of the model during training: early stopping, etc.
• uncertainty quantification using dropout.
• float16, float32, and float64.
• many other useful features: different (weighted) losses, learning rate schedules, metrics, etc.

All the components of DeepXDE are loosely coupled, and thus DeepXDE is well-structured and highly configurable. It is easy to customize DeepXDE to meet new demands.

User guide

user/installation

demos/function demos/pinn_forward demos/pinn_inverse demos/operator user/parallel user/faq

user/research user/cite_deepxde user/team

API reference

If you are looking for information on a specific function, class or method, this part of the documentation is for you.

modules/deepxde modules/deepxde.data modules/deepxde.geometry modules/deepxde.gradients modules/deepxde.icbc modules/deepxde.nn modules/deepxde.nn.jax modules/deepxde.nn.paddle modules/deepxde.nn.pytorch modules/deepxde.nn.tensorflow modules/deepxde.nn.tensorflow_compat_v1 modules/deepxde.optimizers modules/deepxde.utils

Indices and tables

• genindex
• modindex
• search