A Learning-Based Ansatz Satisfying Boundary Conditions in Variational Problems

TL;DR

This work addresses the limitation of the Deep Ritz Method where neural-network test functions do not automatically satisfy boundary conditions. It introduces a boundary-satisfying ansatz $y(x)=B(x)+p(x)N_{net}(x,\theta)$, with $p(x)$ vanishing on the boundary and $B(x)$ encoding boundary data, thereby removing the need for penalty terms. Theoretical justification shows that $p(x)$ times a one-hidden-layer network is dense in the continuous function space, ensuring universal approximation within the Ritz framework. Empirically, the method achieves faster convergence and significantly smaller boundary errors than the traditional Deep Ritz Method across three examples, including a simple 1D action, a 2D Poisson-like problem, and a 3D quantum oscillator, with notably lower boundary deviations and comparable or superior accuracy. Overall, enforcing boundary conditions through the proposed ansatz enhances reliability and efficiency in variational problem solving.

Abstract

Recently, innovative adaptations of the Ritz Method incorporating deep learning have been developed, known as the Deep Ritz Method. This approach employs a neural network as the test function for variational problems. However, the neural network does not inherently satisfy the boundary conditions of the variational problem. To resolve this issue, the Deep Ritz Method introduces a penalty term into the functional of the variational problem, which can lead to misleading results during the optimization process. In this work, an ansatz is proposed that inherently satisfies the boundary conditions of the variational problem. The results demonstrate that the proposed ansatz not only eliminates misleading outcomes but also reduces complexity while maintaining accuracy, showcasing its practical effectiveness in addressing variational problems.

Figures (6)

• Figure 1: Values of $\hat{S}$ with respect to the iteration number of the gradient descent method. These values are presented for three cases: case $N=2$, where $\hat{S}$ is minimized using the ansatz given by \ref{['eq29']}; case $N=2$, where $\hat{S}_{\mathrm{DR}}$ is minimized; and case $N=30$, where $\hat{S}_{\mathrm{DR}}$ is minimized.
• Figure 2: (a) $\hat{y}(u)$ values for the $\theta$ values obtained in the final iteration; (b) absolute error relative to the exact solution given by \ref{['eq31']}. Three cases are considered: the proposed method using the ansatz provided in \ref{['eq27']}, the deep Ritz method with $N=2$, and the deep Ritz method with $N=30$.
• Figure 3: Values of $S$ with respect to the iteration number of the gradient descent method are presented for two cases: the case of $N=10$, where $S$ is minimized using the ansatz provided in \ref{['eq36']}, and the case of $N=10$ with an additional hidden layer, where $S_{\mathrm{DR}}$ is minimized.
• Figure 4: $y(x_1,x_2,\theta)$ values corresponding to the $\theta$ values obtained in the final iteration are shown for (a) the proposed method and (b) the deep Ritz method.
• Figure 5: Absolute errors relative to the exact solution given by \ref{['eq35']}, obtained using (a) the proposed method and (b) the deep Ritz method.

Theorems & Definitions (2)

• proof : Proof of Theorem 1
• proof : Proof of Lemma 1