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

arXiv: https://arxiv.org/abs/2504.05026 · HTML: https://mlsanity.com/paper/2504.05026/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 29 references that are in the map.
- First posted on arXiv 7 Apr 2025.

## Builds on (most cited first, 18 of 18)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Shortcut Learning in Deep Neural Networks](https://mlsanity.com/paper/2004.07780/) (2020)
- [Interpretability Beyond Feature Attribution: Quantitative Testing with Concept Activation Vectors (TCAV)](https://mlsanity.com/paper/1711.11279/) (2017)
- [InceptionTime: Finding AlexNet for Time Series Classification](https://mlsanity.com/paper/1909.04939/) (2019)
- [Opportunities and Challenges in Explainable Artificial Intelligence (XAI): A Survey](https://mlsanity.com/paper/2006.11371/) (2020)
- [On Completeness-aware Concept-Based Explanations in Deep Neural Networks](https://mlsanity.com/paper/1910.07969/) (2019)
- [A Survey on the Explainability of Supervised Machine Learning](https://mlsanity.com/paper/2011.07876/) (2020)
- [Concept Activation Regions: A Generalized Framework For Concept-Based Explanations](https://mlsanity.com/paper/2209.11222/) (2022)
- [ECLAD: Extracting Concepts with Local Aggregated Descriptors](https://mlsanity.com/paper/2206.04531/) (2022)
- [PACE: Posthoc Architecture-Agnostic Concept Extractor for Explaining CNNs](https://mlsanity.com/paper/2108.13828/) (2021)
- [Concept-based explainability for an EEG transformer model](https://mlsanity.com/paper/2307.12745/) (2023)
- [Right on Time: Revising Time Series Models by Constraining their Explanations](https://mlsanity.com/paper/2402.12921/) (2024)
- [Scalable Concept Extraction in Industry 4.0](https://mlsanity.com/paper/2306.03551/) (2023)
- [Scale-Preserving Automatic Concept Extraction (SPACE)](https://mlsanity.com/paper/2308.06022/) (2023)
- [Explaining Deep Neural Networks for Bearing Fault Detection with Vibration Concepts](https://mlsanity.com/paper/2310.11450/) (2023)

## Built on by (most cited first, 1 of 1)

- [Fourier Neural Operators with Least-Squares Readout Refit for Learning Random Obstacle-to-Solution Maps](https://mlsanity.com/paper/2606.29436/) (2026)

## Abstract (arXiv)

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.