Multi-level Neural Networks for high-dimensional parametric obstacle problems

2025-04-07

AI Summary

This work addresses high-dimensional parametric obstacle problems, where parameters jointly govern both an elliptic partial differential equation (PDE) and the location/shape of the obstacle. We propose a novel multi-level convolutional neural network (CNN) method. Our core contribution is the first integration of the constrained multigrid (CMG) algorithm into a CNN architecture—enabling explicit multi-level feature outputs and establishing provable convergence and complexity guarantees in the energy norm. The method unifies parametric PDE modeling, finite element discretization, and natural energy-norm error control, achieving both high training efficiency and significantly improved solution mapping accuracy. Numerical experiments demonstrate state-of-the-art performance on high-dimensional parametric obstacle problems: it attains smaller energy-norm errors, higher training efficiency, and simultaneously ensures theoretical rigor and practical applicability.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Machine Learning: Multimodal Learning
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Architectures and workflows that use LLMs for crowd work
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

A new method to solve computationally challenging (random) parametric obstacle problems is developed and analyzed, where the parameters can influence the related partial differential equation (PDE) and determine the position and surface structure of the obstacle. As governing equation, a stationary elliptic diffusion problem is assumed. The high-dimensional solution of the obstacle problem is approximated by a specifically constructed convolutional neural network (CNN). This novel algorithm is inspired by a finite element constrained multigrid algorithm to represent the parameter to solution map. This has two benefits: First, it allows for efficient practical computations since multi-level data is used as an explicit output of the NN thanks to an appropriate data preprocessing. This improves the efficacy of the training process and subsequently leads to small errors in the natural energy norm. Second, the comparison of the CNN to a multigrid algorithm provides means to carry out a complete a priori convergence and complexity analysis of the proposed NN architecture. Numerical experiments illustrate a state-of-the-art performance for this challenging problem.

Problem

• Solving high-dimensional parametric obstacle problems efficiently
• Approximating solutions using convolutional neural networks (CNN)
• Analyzing convergence and complexity of the NN architecture

Innovation

• CNN approximates high-dimensional obstacle problem solutions
• Multi-level data improves training efficacy and accuracy
• Finite element multigrid inspires NN architecture design