Separable DeepONet: Breaking the Curse of Dimensionality in Physics-Informed Machine Learning

• URL: http://arxiv.org/abs/2407.15887v3
• Date: Tue, 19 Nov 2024 01:30:14 GMT
• Title: Separable DeepONet: Breaking the Curse of Dimensionality in Physics-Informed Machine Learning
• Authors: Luis Mandl, Somdatta Goswami, Lena Lambers, Tim Ricken,
• Abstract summary: 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.
• Score: 0.0
• License:
• Abstract: The deep operator network (DeepONet) is a popular neural operator architecture that has shown promise in solving partial differential equations (PDEs) by using deep neural networks to map between infinite-dimensional function spaces. 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. In this paper, we introduce the Separable DeepONet framework to address these challenges and improve scalability for high-dimensional PDEs. Our approach involves a factorization technique where sub-networks handle individual one-dimensional coordinates, thereby reducing the number of forward passes and the size of the Jacobian matrix. By using forward-mode automatic differentiation, we further optimize the computational cost related to the Jacobian matrix. As a result, our modifications lead to a linear scaling of computational cost with discretization density, making Separable DeepONet suitable for high-dimensional PDEs. We validate the effectiveness of the separable architecture through three benchmark PDE models: the viscous Burgers equation, Biot's consolidation theory, and a parametrized heat equation. In all cases, our proposed framework achieves comparable or improved accuracy while significantly reducing computational time compared to conventional DeepONet. These results demonstrate the potential of Separable DeepONet in efficiently solving complex, high-dimensional PDEs, advancing the field of physics-informed machine learning.

Related papers

• A Deep Learning approach for parametrized and time dependent Partial Differential Equations using Dimensionality Reduction and Neural ODEs [46.685771141109306]
  We propose an autoregressive and data-driven method using the analogy with classical numerical solvers for time-dependent, parametric and (typically) nonlinear PDEs. We show that by leveraging DR we can deliver not only more accurate predictions, but also a considerably lighter and faster Deep Learning model.
  arXiv  Detail & Related papers  (2025-02-12T11:16:15Z)
• Transolver++: An Accurate Neural Solver for PDEs on Million-Scale Geometries [67.63077028746191]
  Transolver++ is a highly parallel and efficient neural solver that can solve PDEs on million-scale geometries. Transolver++ increases the single- GPU input capacity to million-scale points for the first time. It achieves over 20% performance gain in million-scale high-fidelity industrial simulations.
  arXiv  Detail & Related papers  (2025-02-04T15:33:50Z)
• Physics-Informed Latent Neural Operator for Real-time Predictions of Complex Physical Systems [0.0]
  Deep operator network (DeepONet) has shown great promise as a surrogate model for systems governed by partial differential equations (PDEs) This work introduces PI-Latent-NO, a physics-informed latent operator learning framework that overcomes limitations.
  arXiv  Detail & Related papers  (2025-01-14T20:38:30Z)
• Solving Poisson Equations using Neural Walk-on-Spheres [80.1675792181381]
  We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations. We demonstrate the superiority of NWoS in accuracy, speed, and computational costs.
  arXiv  Detail & Related papers  (2024-06-05T17:59:22Z)
• 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)
• LordNet: An Efficient Neural Network for Learning to Solve Parametric Partial Differential Equations without Simulated Data [47.49194807524502]
  We propose LordNet, a tunable and efficient neural network for modeling entanglements. The experiments on solving Poisson's equation and (2D and 3D) Navier-Stokes equation demonstrate that the long-range entanglements can be well modeled by the LordNet.
  arXiv  Detail & Related papers  (2022-06-19T14:41:08Z)
• Physics-constrained Unsupervised Learning of Partial Differential Equations using Meshes [1.066048003460524]
  Graph neural networks show promise in accurately representing irregularly meshed objects and learning their dynamics. In this work, we represent meshes naturally as graphs, process these using Graph Networks, and formulate our physics-based loss to provide an unsupervised learning framework for partial differential equations (PDE) Our framework will enable the application of PDE solvers in interactive settings, such as model-based control of soft-body deformations.
  arXiv  Detail & Related papers  (2022-03-30T19:22:56Z)
• Learning Physics-Informed Neural Networks without Stacked Back-propagation [82.26566759276105]
  We develop a novel approach that can significantly accelerate the training of Physics-Informed Neural Networks. In particular, we parameterize the PDE solution by the Gaussian smoothed model and show that, derived from Stein's Identity, the second-order derivatives can be efficiently calculated without back-propagation. Experimental results show that our proposed method can achieve competitive error compared to standard PINN training but is two orders of magnitude faster.
  arXiv  Detail & Related papers  (2022-02-18T18:07:54Z)
• Solving inverse-PDE problems with physics-aware neural networks [0.0]
  We propose a novel framework to find unknown fields in the context of inverse problems for partial differential equations. We blend the high expressibility of deep neural networks as universal function estimators with the accuracy and reliability of existing numerical algorithms.
  arXiv  Detail & Related papers  (2020-01-10T18:46:50Z)

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.