# Efficient PDE-Constrained optimization under high-dimensional uncertainty using derivative-informed neural operators arXiv: https://arxiv.org/abs/2305.20053 · HTML: https://mlsanity.com/paper/2305.20053/ ## Key facts - Cited by 19 papers in the map (data through 14 Oct 2026). - Builds on 32 of its 74 references that are in the map. - First posted on arXiv 31 May 2023. ## Builds on (most cited first, 30 of 32) - [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020) - [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021) - [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021) - [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020) - [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021) - [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020) - [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021) - [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022) - [Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport](https://mlsanity.com/paper/2204.06684/) (2022) - [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020) - [POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition](https://mlsanity.com/paper/2101.11845/) (2021) - [Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learning](https://mlsanity.com/paper/2206.10745/) (2022) - [Solving PDE-constrained Control Problems Using Operator Learning](https://mlsanity.com/paper/2111.04941/) (2021) - [Derivative-Informed Projected Neural Networks for High-Dimensional Parametric Maps Governed by PDEs](https://mlsanity.com/paper/2011.15110/) (2020) - [Learning to Solve PDE-constrained Inverse Problems with Graph Networks](https://mlsanity.com/paper/2206.00711/) (2022) - [Fast PDE-constrained optimization via self-supervised operator learning](https://mlsanity.com/paper/2110.13297/) (2021) - [Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2210.03008/) (2022) - [Deep neural operators can serve as accurate surrogates for shape optimization: A case study for airfoils](https://mlsanity.com/paper/2302.00807/) (2023) - [Large-scale Bayesian optimal experimental design with derivative-informed projected neural network](https://mlsanity.com/paper/2201.07925/) (2022) - [Bi-level Physics-Informed Neural Networks for PDE Constrained Optimization using Broyden's Hypergradients](https://mlsanity.com/paper/2209.07075/) (2022) - [Learning High-Dimensional Parametric Maps via Reduced Basis Adaptive Residual Networks](https://mlsanity.com/paper/2112.07096/) (2021) - [Optimal design of acoustic metamaterial cloaks under uncertainty](https://mlsanity.com/paper/2007.13252/) (2020) - [Taylor approximation and variance reduction for PDE-constrained optimal control under uncertainty](https://mlsanity.com/paper/1804.04301/) (2018) - [Mean-variance risk-averse optimal control of systems governed by PDEs with random parameter fields using quadratic approximations](https://mlsanity.com/paper/1602.07592/) (2016) - [Certifiable Risk-Based Engineering Design Optimization](https://mlsanity.com/paper/2101.05129/) (2021) - [An efficient, globally convergent method for optimization under uncertainty using adaptive model reduction and sparse grids](https://mlsanity.com/paper/1811.00177/) (2018) - [Taylor approximation for chance constrained optimization problems governed by partial differential equations with high-dimensional random parameters](https://mlsanity.com/paper/2011.09985/) (2020) - [Multilevel Monte Carlo analysis for optimal control of elliptic PDEs with random coefficients](https://mlsanity.com/paper/1602.08943/) (2016) - [A Certified Model Reduction Approach for Robust Parameter Optimization with PDE Constraints](https://mlsanity.com/paper/1703.01613/) (2017) - [Performance Bounds for PDE-Constrained Optimization under Uncertainty](https://mlsanity.com/paper/2110.10269/) (2021) ## Built on by (most cited first, 19 of 19) - [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024) - [Transformers as Neural Operators for Solutions of Differential Equations with Finite Regularity](https://mlsanity.com/paper/2405.19166/) (2024) - [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024) - [Derivative-enhanced Deep Operator Network](https://mlsanity.com/paper/2402.19242/) (2024) - [Derivative-informed neural operator acceleration of geometric MCMC for infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2403.08220/) (2024) - [Residual-Based Error Corrector Operator to Enhance Accuracy and Reliability of Neural Operator Surrogates of Nonlinear Variational Boundary-Value Problems](https://mlsanity.com/paper/2306.12047/) (2023) - [LazyDINO: Fast, scalable, and efficiently amortized Bayesian inversion via structure-exploiting and surrogate-driven measure transport](https://mlsanity.com/paper/2411.12726/) (2024) - [Performance of Neural and Polynomial Operator Surrogates](https://mlsanity.com/paper/2604.00689/) (2026) - [Real-time optimal control of high-dimensional parametrized systems by deep learning-based reduced order models](https://mlsanity.com/paper/2409.05709/) (2024) - [Verification and Validation for Trustworthy Scientific Machine Learning](https://mlsanity.com/paper/2502.15496/) (2025) - [Multifidelity linear regression for scientific machine learning from scarce data](https://mlsanity.com/paper/2403.08627/) (2024) - [Method of Manufactured Learning for Solver-free Training of Neural Operators](https://mlsanity.com/paper/2511.12890/) (2025) - [Fast Unconstrained Optimization via Hessian Averaging and Adaptive Gradient Sampling Methods](https://mlsanity.com/paper/2408.07268/) (2024) - [Sequential Bayesian Optimal Experimental Design in Infinite Dimensions via Policy Gradient Reinforcement Learning](https://mlsanity.com/paper/2601.05868/) (2026) - [Latent feedback control of distributed systems in multiple scenarios through deep learning-based reduced order models](https://mlsanity.com/paper/2412.09942/) (2024) - [A Single-Loop Bilevel Deep Learning Method for Optimal Control of Obstacle Problems](https://mlsanity.com/paper/2601.04120/) (2026) - [Universal Approximation of Nonlinear Operators and Their Derivatives](https://mlsanity.com/paper/2605.15285/) (2026) - [Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solver](https://mlsanity.com/paper/2609.24021/) (2026) - [Tangent Schr"odinger Bridge Matching: Learning Stochastic Transport with Mechanistic Sensitivities](https://mlsanity.com/paper/2610.02906/) (2026) ## Abstract (arXiv) We propose a novel machine learning framework for solving optimization problems governed by large-scale partial differential equations (PDEs) with high-dimensional random parameters. Such optimization under uncertainty (OUU) problems may be computational prohibitive using classical methods, particularly when a large number of samples is needed to evaluate risk measures at every iteration of an optimization algorithm, where each sample requires the solution of an expensive-to-solve PDE. To address this challenge, we propose a new neural operator approximation of the PDE solution operator that has the combined merits of (1) accurate approximation of not only the map from the joint inputs of random parameters and optimization variables to the PDE state, but also its derivative with respect to the optimization variables, (2) efficient construction of the neural network using reduced basis architectures that are scalable to high-dimensional OUU problems, and (3) requiring only a limited number of training data to achieve high accuracy for both the PDE solution and the OUU solution. We refer to such neural operators as multi-input reduced basis derivative informed neural operators (MR-DINOs). We demonstrate the accuracy and efficiency our approach through several numerical experiments, i.e. the risk-averse control of a semilinear elliptic PDE and the steady state Navier--Stokes equations in two and three spatial dimensions, each involving random field inputs. Across the examples, MR-DINOs offer 103--107 × reductions in execution time, and are able to produce OUU solutions of comparable accuracies to those from standard PDE based solutions while being over 10 × more cost-efficient after factoring in the cost of construction.