Inf-Sup neural networks for high-dimensional elliptic PDE problems

TL;DR

Inf-SupNet introduces a mesh-free, unsupervised neural solver for high-dimensional elliptic PDEs by casting the problem into a primal–dual inf–sup (saddle-point) framework and employing two neural networks to represent the primal solution and the Lagrangian multiplier. The authors prove equivalence between the inf–sup formulation and the original PDE, and they decompose the global error into training, sampling, and approximation components with accompanying bounds. Empirically, Inf-SupNet achieves stable and accurate solutions for Poisson, irregular domains, mixed boundary conditions, and both semilinear and nonlinear elliptic problems up to dimension five, using Monte Carlo integration and standard network architectures. The work highlights the practical potential of primal–dual neural PDE solvers while acknowledging optimizer sensitivity and the need for further development to surpass state-of-the-art methods in some settings.

Abstract

Solving high dimensional partial differential equations (PDEs) has historically posed a considerable challenge when utilizing conventional numerical methods, such as those involving domain meshes. Recent advancements in the field have seen the emergence of neural PDE solvers, leveraging deep networks to effectively tackle high dimensional PDE problems. This study introduces Inf-SupNet, a model-based unsupervised learning approach designed to acquire solutions for a specific category of elliptic PDEs. The fundamental concept behind Inf-SupNet involves incorporating the inf-sup formulation of the underlying PDE into the loss function. The analysis reveals that the global solution error can be bounded by the sum of three distinct errors: the numerical integration error, the duality gap of the loss function (training error), and the neural network approximation error for functions within Sobolev spaces. To validate the efficacy of the proposed method, numerical experiments conducted in high dimensions demonstrate its stability and accuracy across various boundary conditions, as well as for both semi-linear and nonlinear PDEs.

Figures (4)

• Figure 1: Numerical results for the Poisson equation with Dirichlet BC. Top left: the true solution $u^*$, top middle: the computed solution $u_\theta$; top right: the Lagrangian multiplier $v_\tau$; bottom left: the difference $u_\theta-u^*$; bottom middle the training loss over iterations; bottom right: the test error (relative $L^2$ error with respect to the true solution) over iterations. The solutions are plotted over $x_2,x_4$ with $x_1=x_3=x_5=0$. The EMAs of the loss and error are plotted at bottom middle and right.
• Figure 2: Numerical results on L-shaped domain. Left: solution computed by InfSupNet; middle: pointwise error with the FEM solution; Right: relative $L^2$ error during training
• Figure 3: Numerical results for mixed boundary conditions. Left: solution computed by InfSupNet; middle: pointwise error with the FEM solution; Right: relative $L^2$ error during training
• Figure 4: Numerical results for semilinear and nonlinear elliptic PDEs with $d=5$. Left: Relative error with iterations for the semilinear PDE; right: Relative error with iterations for the nonlinear PDE

Theorems & Definitions (16)

• theorem 1
• definition 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• theorem 2
• proof