arXiv subjects

Dingcheng Luo

Publications and source records attributed to Dingcheng Luo.

7 recordsLinked to original sources

arXiv2026

Derivative-Informed Fourier Neural Operator: Universal Approximation and Applications to PDE-Constrained Optimization

We present approximation theories and efficient training methods for derivative-informed Fourier neural operators (DIFNOs) with applications to PDE-constrained optimization. A DIFNO is an FNO trained by minimizing its prediction error jointly on output and Fréchet derivative samples of a high-fidelity operator (e.g., a parametric PDE solution operator). As a result, a DIFNO can closely emulate not only the high-fidelity operator's response but also its sensitivities. To motivate the use of DIFNOs instead of conventional FNOs as surrogate models, we show that accurate surrogate-driven PDE-constrained optimization requires accurate surrogate Fréchet derivatives. Then, we establish (i) simultaneous universal approximation of continuously differentiable operators and their Fréchet derivatives by FNOs on compact sets, and (ii) universal approximation of continuously differentiable operators by FNOs in weighted Sobolev spaces with input measures that have unbounded supports. Our theoretical results certify the capability of FNOs for accurate derivative-informed operator learning and for the solution of PDE-constrained optimization problems. Furthermore, we develop efficient training schemes that leverage dimensionality reduction and multi-resolution techniques to significantly reduce memory and computational costs in Fréchet derivative learning. Numerical examples on nonlinear diffusion--reaction, Helmholtz, and Navier--Stokes equations demonstrate that DIFNOs are superior in sample complexity for operator learning and solving infinite-dimensional PDE-constrained inverse problems, achieving high accuracy at low training sample sizes.

cs.LG↗

arXiv2026

Shape Derivative-Informed Neural Operators with Application to Risk-Averse Shape Optimization

Shape optimization under uncertainty (OUU) is computationally intensive for classical PDE-based methods due to the high cost of repeated sampling-based risk evaluation across many uncertainty realizations and varying geometries, while standard neural surrogates often fail to provide accurate and efficient sensitivities for optimization. We introduce Shape-DINO, a derivative-informed neural operator framework for learning PDE solution operators on families of varying geometries, with a particular focus on accelerating PDE-constrained shape OUU. Shape-DINOs encode geometric variability through diffeomorphic mappings to a fixed reference domain and employ a derivative-informed operator learning objective that jointly learns the PDE solution and its Fréchet derivatives with respect to design variables and uncertain parameters, enabling accurate state predictions and reliable gradients for large-scale OUU. We establish a priori error bounds linking surrogate accuracy to optimization error and prove universal approximation results for multi-input reduced basis neural operators in suitable $C^1$ norms. We demonstrate efficiency and scalability on three representative shape OUU problems, including boundary design for a Poisson equation and shape design governed by steady-state Navier-Stokes exterior flows in two and three dimensions. Across these examples, Shape-DINOs produce more reliable optimization results than operator surrogates trained without derivative information. In our examples, Shape-DINOs achieve 3-8 orders-of-magnitude speedups in state and gradient evaluations. Counting training data generation, Shape-DINOs reduce necessary PDE solves by 1-2 orders-of-magnitude compared to a strictly PDE-based approach for a single OUU problem. Moreover, Shape-DINO construction costs can be amortized across many objectives and risk measures, enabling large-scale shape OUU for complex systems.

math.OC↗

arXiv2025

Dimension reduction for derivative-informed operator learning: An analysis of approximation errors

We study the derivative-informed learning of nonlinear operators between infinite-dimensional separable Hilbert spaces by neural networks. Such operators can arise from the solution of partial differential equations (PDEs), and are used in many simulation-based outer-loop tasks in science and engineering, such as PDE-constrained optimization, Bayesian inverse problems, and optimal experimental design. In these settings, the neural network approximations can be used as surrogate models to accelerate the solution of the outer-loop tasks. However, since outer-loop tasks in infinite dimensions often require knowledge of the underlying geometry, the approximation accuracy of the operator's derivatives can also significantly impact the performance of the surrogate model. Motivated by this, we analyze the approximation errors of neural operators in Sobolev norms over infinite-dimensional Gaussian input measures. We focus on the reduced basis neural operator (RBNO), which uses linear encoders and decoders defined on dominant input/output subspaces spanned by reduced sets of orthonormal bases. To this end, we study two methods for generating the bases; principal component analysis (PCA) and derivative-informed subspaces (DIS), which use the dominant eigenvectors of the covariance of the data or the derivatives as the reduced bases, respectively. We then derive bounds for errors arising from both the dimension reduction and the latent neural network approximation, including the sampling errors associated with the empirical estimation of the PCA/DIS. Our analysis is validated on numerical experiments with elliptic PDEs, where our results show that bases informed by the map (i.e., DIS or output PCA) yield accurate reconstructions and generalization errors for both the operator and its derivatives, while input PCA may underperform unless ranks and training sample sizes are sufficiently large.

math.NA↗

arXiv2024

Gaussian mixture Taylor approximations of risk measures constrained by PDEs with Gaussian random field inputs

This work considers the computation of risk measures for quantities of interest governed by PDEs with Gaussian random field parameters using Taylor approximations. While efficient, Taylor approximations are local to the point of expansion, and hence may degrade in accuracy when the variances of the input parameters are large. To address this challenge, we approximate the underlying Gaussian measure by a mixture of Gaussians with reduced variance in a dominant direction of parameter space. Taylor approximations are constructed at the means of each Gaussian mixture component, which are then combined to approximate the risk measures. The formulation is presented in the setting of infinite-dimensional Gaussian random parameters for risk measures including the mean, variance, and conditional value-at-risk. We also provide detailed analysis of the approximations errors arising from two sources: the Gaussian mixture approximation and the Taylor approximations. Numerical experiments are conducted for a semilinear advection-diffusion-reaction equation with a random diffusion coefficient field and for the Helmholtz equation with a random wave speed field. For these examples, the proposed approximation strategy can achieve less than $1\%$ relative error in estimating CVaR with only $\mathcal{O}(10)$ state PDE solves, which is comparable to a standard Monte Carlo estimate with $\mathcal{O}(10^4)$ samples, thus achieving significant reduction in computational cost. The proposed method can therefore serve as a way to rapidly and accurately estimate risk measures under limited computational budgets.

math.NA↗

arXiv2024

Inference of Heterogeneous Material Properties via Infinite-Dimensional Integrated DIC

We present a scalable and efficient framework for the inference of spatially-varying parameters of continuum materials from image observations of their deformations. Our goal is the nondestructive identification of arbitrary damage, defects, anomalies and inclusions without knowledge of their morphology or strength. Since these effects cannot be directly observed, we pose their identification as an inverse problem. Our approach builds on integrated digital image correlation (IDIC, Besnard Hild, Roux, 2006), which poses the image registration and material inference as a monolithic inverse problem, thereby enforcing physical consistency of the image registration using the governing PDE. Existing work on IDIC has focused on low-dimensional parameterizations of materials. In order to accommodate the inference of heterogeneous material propertes that are formally infinite dimensional, we present $\infty$-IDIC, a general formulation of the PDE-constrained coupled image registration and inversion posed directly in the function space setting. This leads to several mathematical and algorithmic challenges arising from the ill-posedness and high dimensionality of the inverse problem. To address ill-posedness, we consider various regularization schemes, namely $H^1$ and total variation for the inference of smooth and sharp features, respectively. To address the computational costs associated with the discretized problem, we use an efficient inexact-Newton CG framework for solving the regularized inverse problem. In numerical experiments, we demonstrate the ability of $\infty$-IDIC to characterize complex, spatially varying Lamé parameter fields of linear elastic and hyperelastic materials. Our method exhibits (i) the ability to recover fine-scale and sharp material features, (ii) mesh-independent convergence performance and hyperparameter selection, (iii) robustness to observational noise.

math.NA↗

arXiv2023

Efficient PDE-Constrained optimization under high-dimensional uncertainty using derivative-informed neural operators

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 $10^{3}$--$10^{7} \times$ reductions in execution time, and are able to produce OUU solutions of comparable accuracies to those from standard PDE based solutions while being over $10 \times$ more cost-efficient after factoring in the cost of construction.

math.OC↗

arXiv2022

Optimal design of chemoepitaxial guideposts for directed self-assembly of block copolymer systems using an inexact-Newton algorithm

Directed self-assembly (DSA) of block-copolymers (BCPs) is one of the most promising developments in the cost-effective production of nanoscale devices. The process makes use of the natural tendency for BCP mixtures to form nanoscale structures upon phase separation. The phase separation can be directed through the use of chemically patterned substrates to promote the formation of morphologies that are essential to the production of semiconductor devices. Moreover, the design of substrate pattern can formulated as an optimization problem for which we seek optimal substrate designs that effectively produce given target morphologies. In this paper, we adopt a phase field model given by a nonlocal Cahn--Hilliard partial differential equation (PDE) based on the minimization of the Ohta--Kawasaki free energy, and present an efficient PDE-constrained optimization framework for the optimal design problem. The design variables are the locations of circular- or strip-shaped guiding posts that are used to model the substrate chemical pattern. To solve the ensuing optimization problem, we propose a variant of an inexact Newton conjugate gradient algorithm tailored to this problem. We demonstrate the effectiveness of our computational strategy on numerical examples that span a range of target morphologies. Owing to our second-order optimizer and fast state solver, the numerical results demonstrate five orders of magnitude reduction in computational cost over previous work. The efficiency of our framework and the fast convergence of our optimization algorithm enable us to rapidly solve the optimal design problem in not only two, but also three spatial dimensions.

math.OC↗