Self-Supervised Learning with Lie Symmetries for Partial Differential Equations

• URL: http://arxiv.org/abs/2307.05432v2
• Date: Wed, 14 Feb 2024 14:59:38 GMT
• Title: Self-Supervised Learning with Lie Symmetries for Partial Differential Equations
• Authors: Gr\'egoire Mialon, Quentin Garrido, Hannah Lawrence, Danyal Rehman, Yann LeCun, Bobak T. Kiani
• Abstract summary: We learn general-purpose representations of PDEs by implementing joint embedding methods for self-supervised learning (SSL) Our representation outperforms baseline approaches to invariant tasks, such as regressing the coefficients of a PDE, while also improving the time-stepping performance of neural solvers. We hope that our proposed methodology will prove useful in the eventual development of general-purpose foundation models for PDEs.
• Score: 25.584036829191902
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Machine learning for differential equations paves the way for computationally efficient alternatives to numerical solvers, with potentially broad impacts in science and engineering. Though current algorithms typically require simulated training data tailored to a given setting, one may instead wish to learn useful information from heterogeneous sources, or from real dynamical systems observations that are messy or incomplete. In this work, we learn general-purpose representations of PDEs from heterogeneous data by implementing joint embedding methods for self-supervised learning (SSL), a framework for unsupervised representation learning that has had notable success in computer vision. Our representation outperforms baseline approaches to invariant tasks, such as regressing the coefficients of a PDE, while also improving the time-stepping performance of neural solvers. We hope that our proposed methodology will prove useful in the eventual development of general-purpose foundation models for PDEs. Code: https://github.com/facebookresearch/SSLForPDEs.

Related papers

• Adversarial Learning for Neural PDE Solvers with Sparse Data [4.226449585713182]
  This study introduces a universal learning strategy for neural network PDEs, named Systematic Model Augmentation for Robust Training. By focusing on challenging and improving the model's weaknesses, SMART reduces generalization error during training under data-scarce conditions.
  arXiv  Detail & Related papers  (2024-09-04T04:18:25Z)
• Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learning [45.78096783448304]
  In this work, seeking data efficiency, we design unsupervised pretraining for PDE operator learning. We mine unlabeled PDE data without simulated solutions, and we pretrain neural operators with physics-inspired reconstruction-based proxy tasks. Our method is highly data-efficient, more generalizable, and even outperforms conventional vision-pretrained models.
  arXiv  Detail & Related papers  (2024-02-24T06:27:33Z)
• Training Deep Surrogate Models with Large Scale Online Learning [48.7576911714538]
  Deep learning algorithms have emerged as a viable alternative for obtaining fast solutions for PDEs. Models are usually trained on synthetic data generated by solvers, stored on disk and read back for training. It proposes an open source online training framework for deep surrogate models.
  arXiv  Detail & Related papers  (2023-06-28T12:02:27Z)
• Meta-Learning for Airflow Simulations with Graph Neural Networks [3.52359746858894]
  We present a meta-learning approach to enhance the performance of learned models on out-of-distribution (OoD) samples. Specifically, we set the airflow simulation in CFD over various airfoils as a meta-learning problem, where each set of examples defined on a single airfoil shape is treated as a separate task. We experimentally demonstrate the efficiency of the proposed approach for improving the OoD generalization performance of learned models.
  arXiv  Detail & Related papers  (2023-06-18T19:25:13Z)
• Model-Based Deep Learning: On the Intersection of Deep Learning and Optimization [101.32332941117271]
  Decision making algorithms are used in a multitude of different applications. Deep learning approaches that use highly parametric architectures tuned from data without relying on mathematical models are becoming increasingly popular. Model-based optimization and data-centric deep learning are often considered to be distinct disciplines.
  arXiv  Detail & Related papers  (2022-05-05T13:40:08Z)
• DEALIO: Data-Efficient Adversarial Learning for Imitation from Observation [57.358212277226315]
  In imitation learning from observation IfO, a learning agent seeks to imitate a demonstrating agent using only observations of the demonstrated behavior without access to the control signals generated by the demonstrator. Recent methods based on adversarial imitation learning have led to state-of-the-art performance on IfO problems, but they typically suffer from high sample complexity due to a reliance on data-inefficient, model-free reinforcement learning algorithms. This issue makes them impractical to deploy in real-world settings, where gathering samples can incur high costs in terms of time, energy, and risk. We propose a more data-efficient IfO algorithm
  arXiv  Detail & Related papers  (2021-03-31T23:46:32Z)
• Model-Based Deep Learning [155.063817656602]
  Signal processing, communications, and control have traditionally relied on classical statistical modeling techniques. Deep neural networks (DNNs) use generic architectures which learn to operate from data, and demonstrate excellent performance. We are interested in hybrid techniques that combine principled mathematical models with data-driven systems to benefit from the advantages of both approaches.
  arXiv  Detail & Related papers  (2020-12-15T16:29:49Z)
• Large-scale Neural Solvers for Partial Differential Equations [48.7576911714538]
  Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. Recent numerical solvers require manual discretization of the underlying equation as well as sophisticated, tailored code for distributed computing. We examine the applicability of continuous, mesh-free neural solvers for partial differential equations, physics-informed neural networks (PINNs) We discuss the accuracy of GatedPINN with respect to analytical solutions -- as well as state-of-the-art numerical solvers, such as spectral solvers.
  arXiv  Detail & Related papers  (2020-09-08T13:26:51Z)
• Learning Similarity Metrics for Numerical Simulations [29.39625644221578]
  We propose a neural network-based approach that computes a stable and generalizing metric (LSiM) to compare data from a variety of numerical simulation sources. Our method employs a Siamese network architecture that is motivated by the mathematical properties of a metric.
  arXiv  Detail & Related papers  (2020-02-18T20:11:15Z)

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.