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Neural Parameter Regression for Explicit Representations of PDE Solution Operators

Konrad Mundinger, Max Zimmer, Sebastian Pokutta

TL;DR

Neural Parameter Regression (NPR) addresses learning PDE solution operators across varying initial conditions by combining Hypernetworks with Physics-Informed learning. It parameterizes the output network with a low-rank hidden-weight representation and enforces the initial condition through a reparameterization, enabling efficient, self-supervised operator learning. On 1D heat and Burgers IBVPs, NPR achieves competitive accuracy with Physics-Informed DeepONets and demonstrates strong adaptability to out-of-distribution initial conditions via rapid fine-tuning. This approach offers a scalable path for rapid inference and adaptation in operator learning, with potential for real-time design and control applications, while highlighting limitations in high-dimensional domains and more challenging solution features.

Abstract

We introduce Neural Parameter Regression (NPR), a novel framework specifically developed for learning solution operators in Partial Differential Equations (PDEs). Tailored for operator learning, this approach surpasses traditional DeepONets (Lu et al., 2021) by employing Physics-Informed Neural Network (PINN, Raissi et al., 2019) techniques to regress Neural Network (NN) parameters. By parametrizing each solution based on specific initial conditions, it effectively approximates a mapping between function spaces. Our method enhances parameter efficiency by incorporating low-rank matrices, thereby boosting computational efficiency and scalability. The framework shows remarkable adaptability to new initial and boundary conditions, allowing for rapid fine-tuning and inference, even in cases of out-of-distribution examples.

Neural Parameter Regression for Explicit Representations of PDE Solution Operators

TL;DR

Neural Parameter Regression (NPR) addresses learning PDE solution operators across varying initial conditions by combining Hypernetworks with Physics-Informed learning. It parameterizes the output network with a low-rank hidden-weight representation and enforces the initial condition through a reparameterization, enabling efficient, self-supervised operator learning. On 1D heat and Burgers IBVPs, NPR achieves competitive accuracy with Physics-Informed DeepONets and demonstrates strong adaptability to out-of-distribution initial conditions via rapid fine-tuning. This approach offers a scalable path for rapid inference and adaptation in operator learning, with potential for real-time design and control applications, while highlighting limitations in high-dimensional domains and more challenging solution features.

Abstract

We introduce Neural Parameter Regression (NPR), a novel framework specifically developed for learning solution operators in Partial Differential Equations (PDEs). Tailored for operator learning, this approach surpasses traditional DeepONets (Lu et al., 2021) by employing Physics-Informed Neural Network (PINN, Raissi et al., 2019) techniques to regress Neural Network (NN) parameters. By parametrizing each solution based on specific initial conditions, it effectively approximates a mapping between function spaces. Our method enhances parameter efficiency by incorporating low-rank matrices, thereby boosting computational efficiency and scalability. The framework shows remarkable adaptability to new initial and boundary conditions, allowing for rapid fine-tuning and inference, even in cases of out-of-distribution examples.
Paper Structure (20 sections, 15 equations, 4 figures, 2 tables, 1 algorithm)

This paper contains 20 sections, 15 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methods
  3. Experimental Setup and Results
  4. Heat equation.
  5. Burgers equation.
  6. Conclusion
  7. Appendix
  8. Related Work
  9. Partial differential equations
  10. Heat equation
  11. Burgers equation
  12. Hyperparameters
  13. Evaluation details
  14. Physics-Informed Neural Networks
  15. (Physics-Informed) DeepONets
  16. ...and 5 more sections

Figures (4)

  • Figure 1: The proposed architecture: The Hypernetwork $H_\Phi$ maps an initial condition to another function parametrized by $T_{H_\Phi}$, hence approximating a mapping between function spaces.
  • Figure 2: The heat equation: The first and second row correspond to $u_0(x) = 0.5 \sin(4 \pi x) + \cos(2 \pi x)+ 0.3 \cos(6 \pi x) + 0.8$ and the $u_0(x) = \sum_{n=1}^3 (\sin(n \pi x) + \cos(n \pi x)) + 1$, respectively. The columns shows the reference solutions, the results of and the absolute differences, respectively.
  • Figure 3: The heat equation with the out-of-distribution condition $u_0(x) = 5x + 3 \sin(4 \pi x)$. We plot the reference solution (left), the absolute difference to the reference solution before fine-tuning (middle) and after fine-tuning (right).
  • Figure 4: The Burgers equation: The first row shows the results for the initial condition $u_0(x) = -0.9x + 1.1$ and the second row for $u_0(x) = -0.2x + 1.8$.