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Improving physics-informed DeepONets with hard constraints

Rüdiger Brecht, Dmytro R. Popovych, Alex Bihlo, Roman O. Popovych

TL;DR

This study proposes to improve current physics-informed deep learning strategies such that initial and/or boundary conditions do not need to be learned and are represented exactly in the predicted solution.

Abstract

Current physics-informed (standard or deep operator) neural networks still rely on accurately learning the initial and/or boundary conditions of the system of differential equations they are solving. In contrast, standard numerical methods involve such conditions in computations without needing to learn them. In this study, we propose to improve current physics-informed deep learning strategies such that initial and/or boundary conditions do not need to be learned and are represented exactly in the predicted solution. Moreover, this method guarantees that when a deep operator network is applied multiple times to time-step a solution of an initial value problem, the resulting function is at least continuous.

Improving physics-informed DeepONets with hard constraints

TL;DR

This study proposes to improve current physics-informed deep learning strategies such that initial and/or boundary conditions do not need to be learned and are represented exactly in the predicted solution.

Abstract

Current physics-informed (standard or deep operator) neural networks still rely on accurately learning the initial and/or boundary conditions of the system of differential equations they are solving. In contrast, standard numerical methods involve such conditions in computations without needing to learn them. In this study, we propose to improve current physics-informed deep learning strategies such that initial and/or boundary conditions do not need to be learned and are represented exactly in the predicted solution. Moreover, this method guarantees that when a deep operator network is applied multiple times to time-step a solution of an initial value problem, the resulting function is at least continuous.
Paper Structure (12 sections, 57 equations, 8 figures, 6 tables)

This paper contains 12 sections, 57 equations, 8 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Solving differential equations with neural networks
  3. Physics-informed neural networks
  4. Deep operator neural networks with soft constraints (sONets)
  5. Deep operator neural networks with hard constraints (hONets)
  6. Results
  7. Gravity pendulum with damping
  8. The Lorenz--1963 model
  9. One-dimensional Poisson equation
  10. One-dimensional linear wave equation
  11. Korteweg--de Vries equation
  12. Conclusion

Figures (8)

  • Figure 1: Overview of the used neural network architecture. Here, multilayer perceptron is abbreviated by MLP.
  • Figure 2: The evolution of pointwise error of 10 ahONet1 training runs in the course of time stepping $100$ times corresponding to the time interval $[0,100]$.
  • Figure 3: The evolution of absolute error averaged over 10 runs for the damped gravity pendulum. Left: Errors over a single time-step, i.e., the training interval $[0,1]$ for the DeepOnets. Right: Errors over a total of one hundred time steps, using the time-stepping capabilities of DeepOnets.
  • Figure 4: Numerical results for the Lorenz--1963 system. Left: Trajectories of the Lorenz--1963 system for the sONets and hONets. Right: Zoomed in region between time steps highlighting the discontinuous/continuous nature of the obtained global solution for both methods.
  • Figure 5: Absolute error of the hard- and soft-constrained solutions for the one-dimensional Poisson equation averaged over 100 random samples of $(a,b)$ and 10 runs.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Remark 1
  • Example 1: Dirichlet boundary condition in 1D
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6