# Deep neural network surrogates for non-smooth quantities of interest in shape uncertainty quantification

arXiv: https://arxiv.org/abs/2101.07023 · HTML: https://mlsanity.com/paper/2101.07023/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 76 references that are in the map.
- First posted on arXiv 18 Jan 2021.

## Builds on (most cited first, 25 of 25)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications](https://mlsanity.com/paper/1908.10407/) (2019)
- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [The phase diagram of approximation rates for deep neural networks](https://mlsanity.com/paper/1906.09477/) (2019)
- [Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification](https://mlsanity.com/paper/1802.00850/) (2018)
- [A Theoretical Analysis of Deep Neural Networks and Parametric PDEs](https://mlsanity.com/paper/1904.00377/) (2019)
- [Deep Neural Networks Learn Non-Smooth Functions Effectively](https://mlsanity.com/paper/1802.04474/) (2018)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)
- [Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations](https://mlsanity.com/paper/1809.03062/) (2018)
- [Nonlinear Approximation and (Deep) ReLU Networks](https://mlsanity.com/paper/1905.02199/) (2019)
- [Data-driven discovery of PDEs in complex datasets](https://mlsanity.com/paper/1808.10788/) (2018)
- [Deep learning observables in computational fluid dynamics](https://mlsanity.com/paper/1903.03040/) (2019)
- [Physics Informed Neural Networks for Simulating Radiative Transfer](https://mlsanity.com/paper/2009.13291/) (2020)
- [A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients](https://mlsanity.com/paper/1809.07321/) (2018)
- [Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks](https://mlsanity.com/paper/2004.12131/) (2020)

## Built on by (most cited first, 4 of 4)

- [Hyper-parameter tuning of physics-informed neural networks: Application to Helmholtz problems](https://mlsanity.com/paper/2205.06704/) (2022)
- [Learning smooth functions in high dimensions: from sparse polynomials to deep neural networks](https://mlsanity.com/paper/2404.03761/) (2024)
- [Physics-informed neural networks for operator equations with stochastic data](https://mlsanity.com/paper/2211.10344/) (2022)
- [Generalization Bounds for Equivariant Networks on Markov Data](https://mlsanity.com/paper/2503.00292/) (2025)

## Abstract (arXiv)

We consider the point evaluation of the solution to interface problems with geometric uncertainties, where the uncertainty in the obstacle is described by a high-dimensional parameter boldsymboly∈[-1,1]d, d∈mathbbN. We focus in particular on an elliptic interface problem and a Helmholtz transmission problem. Point values of the solution in the physical domain depend in general non-smoothly on the high-dimensional parameter, posing a challenge when one is interested in building surrogates. Indeed, high-order methods show poor convergence rates, while methods which are able to track discontinuities usually suffer from the so-called curse of dimensionality. For this reason, in this work we propose to build surrogates for point evaluation using deep neural networks. We provide a theoretical justification for why we expect neural networks to provide good surrogates. Furthermore, we present extensive numerical experiments showing their good performance in practice. We observe in particular that neural networks do not suffer from the curse of dimensionality, and we study the dependence of the error on the number of point evaluations (that is, the number of discontinuities in the parameter space), as well as on several modeling parameters, such as the contrast between the two materials and, for the Helmholtz transmission problem, the wavenumber.