Derivative-Informed Projected Neural Networks for High-Dimensional Parametric Maps Governed by PDEs

• URL: http://arxiv.org/abs/2011.15110v2
• Date: Tue, 16 Mar 2021 22:08:58 GMT
• Title: Derivative-Informed Projected Neural Networks for High-Dimensional Parametric Maps Governed by PDEs
• Authors: Thomas O'Leary-Roseberry, Umberto Villa, Peng Chen, and Omar Ghattas
• Abstract summary: We construct surrogates for high-dimensional PDE-governed parametric maps in the form of projected neural networks. We demonstrate that the proposed projected neural network achieves greater accuracy than a full neural network.
• Score: 6.178864935410097
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Many-query problems, arising from uncertainty quantification, Bayesian inversion, Bayesian optimal experimental design, and optimization under uncertainty-require numerous evaluations of a parameter-to-output map. These evaluations become prohibitive if this parametric map is high-dimensional and involves expensive solution of partial differential equations (PDEs). To tackle this challenge, we propose to construct surrogates for high-dimensional PDE-governed parametric maps in the form of projected neural networks that parsimoniously capture the geometry and intrinsic low-dimensionality of these maps. Specifically, we compute Jacobians of these PDE-based maps, and project the high-dimensional parameters onto a low-dimensional derivative-informed active subspace; we also project the possibly high-dimensional outputs onto their principal subspace. This exploits the fact that many high-dimensional PDE-governed parametric maps can be well-approximated in low-dimensional parameter and output subspace. We use the projection basis vectors in the active subspace as well as the principal output subspace to construct the weights for the first and last layers of the neural network, respectively. This frees us to train the weights in only the low-dimensional layers of the neural network. The architecture of the resulting neural network captures to first order, the low-dimensional structure and geometry of the parametric map. We demonstrate that the proposed projected neural network achieves greater generalization accuracy than a full neural network, especially in the limited training data regime afforded by expensive PDE-based parametric maps. Moreover, we show that the number of degrees of freedom of the inner layers of the projected network is independent of the parameter and output dimensions, and high accuracy can be achieved with weight dimension independent of the discretization dimension.

Related papers

• Separable DeepONet: Breaking the Curse of Dimensionality in Physics-Informed Machine Learning [0.0]
  In the absence of labeled datasets, we utilize the PDE residual loss to learn the physical system, an approach known as physics-informed DeepONet. This method faces significant computational challenges, primarily due to the curse of dimensionality, as the computational cost increases exponentially with finer discretization. We introduce the Separable DeepONet framework to address these challenges and improve scalability for high-dimensional PDEs.
  arXiv  Detail & Related papers  (2024-07-21T16:33:56Z)
• ParaPoint: Learning Global Free-Boundary Surface Parameterization of 3D Point Clouds [52.03819676074455]
  ParaPoint is an unsupervised neural learning pipeline for achieving global free-boundary surface parameterization. This work makes the first attempt to investigate neural point cloud parameterization that pursues both global mappings and free boundaries.
  arXiv  Detail & Related papers  (2024-03-15T14:35:05Z)
• Sample Complexity of Neural Policy Mirror Descent for Policy Optimization on Low-Dimensional Manifolds [75.51968172401394]
  We study the sample complexity of the neural policy mirror descent (NPMD) algorithm with deep convolutional neural networks (CNN) In each iteration of NPMD, both the value function and the policy can be well approximated by CNNs. We show that NPMD can leverage the low-dimensional structure of state space to escape from the curse of dimensionality.
  arXiv  Detail & Related papers  (2023-09-25T07:31:22Z)
• Tackling the Curse of Dimensionality with Physics-Informed Neural Networks [24.86574584293979]
  We develop a new method of scaling up physics-informed neural networks (PINNs) to solve arbitrary high-dimensional PDEs. We demonstrate in various tests that the proposed method can solve many notoriously hard high-dimensional PDEs.
  arXiv  Detail & Related papers  (2023-07-23T12:18:12Z)
• Data-Free Learning of Reduced-Order Kinematics [54.85157881323157]
  We produce a low-dimensional map whose image parameterizes a diverse yet low-energy submanifold of configurations. We represent subspaces as neural networks that map a low-dimensional latent vector to the full configuration space. This formulation is effective across a very general range of physical systems.
  arXiv  Detail & Related papers  (2023-05-05T20:53:36Z)
• Multilevel CNNs for Parametric PDEs [0.0]
  We combine concepts from multilevel solvers for partial differential equations with neural network based deep learning. An in-depth theoretical analysis shows that the proposed architecture is able to approximate multigrid V-cycles to arbitrary precision. We find substantial improvements over state-of-the-art deep learning-based solvers.
  arXiv  Detail & Related papers  (2023-04-01T21:11:05Z)
• Solving High-Dimensional PDEs with Latent Spectral Models [74.1011309005488]
  We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space. LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks.
  arXiv  Detail & Related papers  (2023-01-30T04:58:40Z)
• Lie Point Symmetry Data Augmentation for Neural PDE Solvers [69.72427135610106]
  We present a method, which can partially alleviate this problem, by improving neural PDE solver sample complexity. In the context of PDEs, it turns out that we are able to quantitatively derive an exhaustive list of data transformations. We show how it can easily be deployed to improve neural PDE solver sample complexity by an order of magnitude.
  arXiv  Detail & Related papers  (2022-02-15T18:43:17Z)
• Parametric Complexity Bounds for Approximating PDEs with Neural Networks [41.46028070204925]
  We prove that when a PDE's coefficients are representable by small neural networks, the parameters required to approximate its solution scalely with the input $d$ are proportional to the parameter counts of the neural networks. Our proof is based on constructing a neural network which simulates gradient descent in an appropriate space which converges to the solution of the PDE.
  arXiv  Detail & Related papers  (2021-03-03T02:42:57Z)
• Projected Wasserstein gradient descent for high-dimensional Bayesian inference [8.750791391857264]
  We propose a projected Wasserstein gradient descent method (pWGD) for high-dimensional Bayesian inference problems. We overcome this challenge by exploiting the intrinsic low-rank structure in the difference between the posterior and prior distributions.
  arXiv  Detail & Related papers  (2021-02-12T05:12:05Z)
• Augmented Sliced Wasserstein Distances [55.028065567756066]
  We propose a new family of distance metrics, called augmented sliced Wasserstein distances (ASWDs) ASWDs are constructed by first mapping samples to higher-dimensional hypersurfaces parameterized by neural networks. Numerical results demonstrate that the ASWD significantly outperforms other Wasserstein variants for both synthetic and real-world problems.
  arXiv  Detail & Related papers  (2020-06-15T23:00:08Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.