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DeepXDE: A deep learning library for solving differential equations
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Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation. The PINN algorithm is simple, and it can be applied to different types of PDEs, including integro-differential equations, fractional PDEs, and stochastic PDEs. Moreover, from the implementation point of view, PINNs solve inverse problems as easily as forward problems. We propose a new residual-based adaptive refinement (RAR) method to improve the training efficiency of PINNs. For pedagogical reasons, we compare the PINN algorithm to a standard finite element method. We also present a Python library for PINNs, DeepXDE, which is designed to serve both as an education tool to be used in the classroom as well as a research tool for solving problems in computational science and engineering. Specifically, DeepXDE can solve forward problems given initial and boundary conditions, as well as inverse problems given some extra measurements. DeepXDE supports complex-geometry domains based on the technique of constructive solid geometry, and enables the user code to be compact, resembling closely the mathematical formulation. We introduce the usage of DeepXDE and its customizability, and we also demonstrate the capability of PINNs and the user-friendliness of DeepXDE for five different examples. More broadly, DeepXDE contributes to the more rapid development of the emerging Scientific Machine Learning field.
Forward citations
Cited by 6 Pith papers
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Estimation of Hemodynamic Parameters via Physics Informed Neural Networks including Hematocrit Dependent Rheology
PINNs reconstruct smooth velocity and pressure fields from synthetic 4D-flow MRI data for anemic to polycythemic blood, and combining the PINN velocity field with the vWERP estimator gives the most accurate pressure drops.
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Convergence of Physics-Informed Neural Networks for Fully Nonlinear PDE's
Minimizers of the PINN loss are claimed to converge uniformly to the unique viscosity solution of degenerate elliptic fully nonlinear PDEs, under zero-loss capacity and sample-Hölder assumptions.
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Geometry-aware PINNs for Turbulent Flow Prediction
An SDF-embedded RANS-PINN predicts turbulent flow fields for unseen NACA 4-digit airfoil geometries and Reynolds numbers with a few percent normalized error.
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Hybrid Adaptive Modeling in Process Monitoring: Leveraging Sequence Encoders and Physics-Informed Neural Networks
This paper introduces a physics-informed neural network that uses Deep Sets to encode sensor data, allowing one model to adapt to new parameters and boundary conditions without retraining, with tests on a chaotic ODE,...
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How Learnable Grids Recover Fine Detail in Low Dimensions: A Neural Tangent Kernel Analysis of Multigrid Parametric Encodings
The paper claims to prove that multigrid parametric encodings raise the NTK spectrum through their learnable grid, but the proof depends on an invalid additive kernel decomposition.
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Mass-Conserving Physics-Informed Neural Networks For The One-Dimensional Advection-Diffusion Equation
Adding a soft mass-conservation penalty to PINNs for the 1D advection-diffusion equation reduces long-term relative L2 error by 9–67× and mass error by 15–215× compared to vanilla PINNs across Peclet numbers 0.01–20.
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