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Solving parametric PDE problems with artificial neural networks

Yuehaw Khoo, Jianfeng Lu, Lexing Ying

TL;DR

This work tackles the challenge of high-dimensional uncertainty quantification in PDEs by learning a neural-network surrogate that maps high-dimensional coefficient fields $a$ to the PDE-derived quantity $f(a)$. By viewing the PDE solution as a time-evolution process, the authors justify that a convolutional neural network can represent $f(a)$ with controllable error, and provide a constructive, theory-grounded framework. They demonstrate the approach on two parametric PDE problems—the effective conductance of inhomogeneous elliptic media and the ground-state energy of the NLSE with random potential—showing that simple CNNs can achieve around $10^{-3}$ accuracy. The combination of a theoretical representability result and practical numerical experiments supports the viability of NN-based model reduction for PDEs with uncertainty, suggesting a path toward efficient surrogate models and gradient-based design under uncertainty.

Abstract

The curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modeled into the equations as random coefficients. However, very often the variability of physical quantities derived from a PDE can be captured by a few features on the space of the coefficient fields. Based on such an observation, we propose using a neural-network (NN) based method to parameterize the physical quantity of interest as a function of input coefficients. The representability of such quantity using a neural-network can be justified by viewing the neural-network as performing time evolution to find the solutions to the PDE. We further demonstrate the simplicity and accuracy of the approach through notable examples of PDEs in engineering and physics.

Solving parametric PDE problems with artificial neural networks

TL;DR

This work tackles the challenge of high-dimensional uncertainty quantification in PDEs by learning a neural-network surrogate that maps high-dimensional coefficient fields to the PDE-derived quantity . By viewing the PDE solution as a time-evolution process, the authors justify that a convolutional neural network can represent with controllable error, and provide a constructive, theory-grounded framework. They demonstrate the approach on two parametric PDE problems—the effective conductance of inhomogeneous elliptic media and the ground-state energy of the NLSE with random potential—showing that simple CNNs can achieve around accuracy. The combination of a theoretical representability result and practical numerical experiments supports the viability of NN-based model reduction for PDEs with uncertainty, suggesting a path toward efficient surrogate models and gradient-based design under uncertainty.

Abstract

The curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modeled into the equations as random coefficients. However, very often the variability of physical quantities derived from a PDE can be captured by a few features on the space of the coefficient fields. Based on such an observation, we propose using a neural-network (NN) based method to parameterize the physical quantity of interest as a function of input coefficients. The representability of such quantity using a neural-network can be justified by viewing the neural-network as performing time evolution to find the solutions to the PDE. We further demonstrate the simplicity and accuracy of the approach through notable examples of PDEs in engineering and physics.

Paper Structure

This paper contains 12 sections, 3 theorems, 35 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Two examples of parametric PDE problems
  3. Effective coefficients for inhomogeneous elliptic equation
  4. NLSE with inhomogeneous background potential
  5. Theoretical justification of deep neural-network representation
  6. Proposed network architecture and numerical results
  7. Architecture
  8. Implementation
  9. Numerical examples
  10. Effective conductance
  11. Ground state energy of NLSE
  12. Conclusion

Key Result

Theorem 1

Fix an error tolerance $\epsilon > 0$, there exists a neural-network $h_\theta(\cdot)$ with $O(n^d)$ hidden nodes per-layer and $O((\frac{\lambda_1}{\lambda_0}+1)\frac{n^{2}}{\epsilon})$ layers such that for any $a\in\mathcal{A}=\{a\in\mathbb{R}^{n^d} \mid a_i \in [\lambda_0, \lambda_1],\,\forall i\

Figures (4)

  • Figure 1: Construction of the NN in the proof of Theorem \ref{['theorem: effective conductance']}. The NN takes the coefficient field $a$ as an input and the convolutional and local nonlinearity layers are used to map from $u^m, a$ to $u^{m+1},a$, $m=0,M-1$. At $u_M$, local convolutions and nonlinearity are used to obtained $\mathcal{E}(u^M;a)$.
  • Figure 2: Single convolutional layer neural network for representing translational invariant function.
  • Figure 3: Neural-network architecture for approximating $A_{\mathrm{eff}}(a)$ in the 1D case. Although the layers in third stage are essentially densely-connected layers, we still identify them as convolution layers to reflect the symmetry between the first and third stages.
  • Figure 4: The first stage's output of the neural-network in Fig. \ref{['figure: homo1D']} fitted by $\beta_1/x + \beta_2$. The training result agrees well with the analytical structure of the solution to the 1D effective conductance.

Theorems & Definitions (5)

  • Theorem 1
  • Lemma 1
  • proof
  • Theorem 2
  • proof