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Zero Coordinate Shift: Whetted Automatic Differentiation for Physics-informed Operator Learning

arXiv 2311.00860 ↗First posted 1 Nov 2023cs.LG cs.AI cs.NAcited by 0

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Abstract (arXiv)

Automatic differentiation (AD) is a critical step in physics-informed machine learning, required for computing the high-order derivatives of network output w.r.t. coordinates of collocation points. In this paper, we present a novel and lightweight algorithm to conduct AD for physics-informed operator learning, which we call the trick of Zero Coordinate Shift (ZCS). Instead of making all sampled coordinates as leaf variables, ZCS introduces only one scalar-valued leaf variable for each spatial or temporal dimension, simplifying the wanted derivatives from "many-roots-many-leaves" to "one-root-many-leaves" whereby reverse-mode AD becomes directly utilisable. It has led to an outstanding performance leap by avoiding the duplication of the computational graph along the dimension of functions (physical parameters). ZCS is easy to implement with current deep learning libraries; our own implementation is achieved by extending the DeepXDE package. We carry out a comprehensive benchmark analysis and several case studies, training physics-informed DeepONets to solve partial differential equations (PDEs) without data. The results show that ZCS has persistently reduced GPU memory consumption and wall time for training by an order of magnitude, and such reduction factor scales with the number of functions. As a low-level optimisation technique, ZCS imposes no restrictions on data, physics (PDE) or network architecture and does not compromise training results from any aspect.

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 26 of its 41 references that are in the map. First posted on arXiv 1 Nov 2023.

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References · 26

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  4. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  5. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  6. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
  7. 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 220
  8. 2020PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domaincited by 173
  9. 2021Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equationscited by 142
  10. 2021Exact imposition of boundary conditions with distance functions in physics-informed deep neural networkscited by 135
  11. 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 133
  12. 2017A unified deep artificial neural network approach to partial differential equations in complex geometriescited by 122
14 more
  1. 2021PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEscited by 97
  2. 2020A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networkscited by 86
  3. 2020Learning Differential Equations that are Easy to Solvecited by 81
  4. 2020Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domainscited by 78
  5. 2021Understanding Pooling in Graph Neural Networkscited by 69
  6. 2023Separable Physics-Informed Neural Networkscited by 64
  7. 2019A Phase Shift Deep Neural Network for High Frequency Approximation and Wave Problemscited by 34
  8. 2022Accelerated Training of Physics-Informed Neural Networks (PINNs) using Meshless Discretizationscited by 26
  9. 2020Linearly Constrained Neural Networkscited by 18
  10. 2023Fuzzing Automatic Differentiation in Deep-Learning Librariescited by 16
  11. 2022JAX-DIPS: Neural bootstrapping of finite discretization methods and application to elliptic problems with discontinuitiescited by 7
  12. 2022Replacing Automatic Differentiation by Sobolev Cubatures fastens Physics Informed Neural Nets and strengthens their Approximation Powercited by 4
  13. 2023Software-based Automatic Differentiation is Flawedcited by 3
  14. 2022Neuro-symbolic partial differential equation solvercited by 1

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