A semigroup method for high dimensional elliptic PDEs and eigenvalue problems based on neural networks

Li, Haoya; Ying, Lexing

doi:10.1016/j.jcp.2022.110939

Cited by 7 publications

(4 citation statements)

References 10 publications

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“…However, deep learning has demonstrated remarkable flexibility and adaptivity in approximating highdimensional functions, which indeed has led to significant advances in computer vision and natural language processing. Recently, a series of works (Han, Jentzen, and E 2018;Yu et al 2018;Karniadakis et al 2021;Khoo, Lu, and Ying 2021;Long et al 2018;Zang et al 2020;Kovachki et al 2021;Lu, Jin, and Karniadakis 2019;Li et al 2022;Rotskoff, Mitchell, and Vanden-Eijnden 2022) that assume the sampled data are independent and identically distributed. This i.i.d.…”

Section: Related Workmentioning

confidence: 99%

Statistical Spatially Inhomogeneous Diffusion Inference

Ren,

Lu,

Ying

et al. 2024

AAAI

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Inferring a diffusion equation from discretely observed measurements is a statistical challenge of significant importance in a variety of fields, from single-molecule tracking in biophysical systems to modeling financial instruments. Assuming that the underlying dynamical process obeys a d-dimensional stochastic differential equation of the form dx_t = b(x_t)dt + \Sigma(x_t)dw_t, we propose neural network-based estimators of both the drift b and the spatially-inhomogeneous diffusion tensor D = \Sigma\Sigma^T/2 and provide statistical convergence guarantees when b and D are s-Hölder continuous. Notably, our bound aligns with the minimax optimal rate N^{-\frac{2s}{2s+d}} for nonparametric function estimation even in the presence of correlation within observational data, which necessitates careful handling when establishing fast-rate generalization bounds. Our theoretical results are bolstered by numerical experiments demonstrating accurate inference of spatially-inhomogeneous diffusion tensors.

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Section: Related Workmentioning

confidence: 99%

Statistical Spatially Inhomogeneous Diffusion Inference

Ren,

Lu,

Ying

et al. 2024

AAAI

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“…The Deep Ritz method (DRM) is one of the most renowned approaches in the field of elliptic equations, capable of solving both the equations and the eigenvalue problems [9,12,14,17,19,25,27,28,30]. In this article, we present its application in nonlinear elliptic equations and provide a convergent analysis.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Deep Ritz Methods for Semilinear Elliptic Equations

Mo Chen,

Yuling Jiao,

Xiliang Lu

et al. 2024

NMTMA

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In this paper, we propose a method for solving semilinear elliptical equations using a ResNet with ReLU 2 activations. Firstly, we present a comprehensive formulation based on the penalized variational form of the elliptical equations. We then apply the Deep Ritz Method, which works for a wide range of equations. We obtain an upper bound on the errors between the acquired solutions and the true solutions in terms of the depth D, width W of the ReLU 2 ResNet, and the number of training samples n. Our simulation results demonstrate that our method can effectively overcome the curse of dimensionality and validate the theoretical results.

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“…Recently, certain artificial neural networks (ANNs) based approximation methods for PDEs have been proposed and various numerical simulations for such methods suggest (cf., e.g., [9,11,13,14,19,21,22,24,29,30,32,35,38,40,41,46,47,[49][50][51][55][56][57]59,60] and the references mentioned therein) that deep ANNs might have the capacity to indeed overcome the curse of dimensionality in the sense that the number of real parameters used to describe the approximating deep ANNs grows at most polynomially in both the PDE dimension d ∈ N = {1, 2, . .…”

Section: Introductionmentioning

confidence: 99%