Mechanistic PDE Networks for Discovery of Governing Equations

• URL: http://arxiv.org/abs/2502.18377v1
• Date: Tue, 25 Feb 2025 17:21:44 GMT
• Title: Mechanistic PDE Networks for Discovery of Governing Equations
• Authors: Adeel Pervez, Efstratios Gavves, Francesco Locatello,
• Abstract summary: We present Mechanistic PDE Networks, a model for discovery of partial differential equations from data.<n>The represented PDEs are then solved and decoded for specific tasks.<n>We develop a native, GPU-capable, parallel, sparse, and differentiable multigrid solver specialized for linear partial differential equations.
• Score: 52.492158106791365
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: We present Mechanistic PDE Networks -- a model for discovery of governing partial differential equations from data. Mechanistic PDE Networks represent spatiotemporal data as space-time dependent linear partial differential equations in neural network hidden representations. The represented PDEs are then solved and decoded for specific tasks. The learned PDE representations naturally express the spatiotemporal dynamics in data in neural network hidden space, enabling increased power for dynamical modeling. Solving the PDE representations in a compute and memory-efficient way, however, is a significant challenge. We develop a native, GPU-capable, parallel, sparse, and differentiable multigrid solver specialized for linear partial differential equations that acts as a module in Mechanistic PDE Networks. Leveraging the PDE solver, we propose a discovery architecture that can discover nonlinear PDEs in complex settings while also being robust to noise. We validate PDE discovery on a number of PDEs, including reaction-diffusion and Navier-Stokes equations.

Related papers

• Advancing Generalization in PINNs through Latent-Space Representations [71.86401914779019]
  Physics-informed neural networks (PINNs) have made significant strides in modeling dynamical systems governed by partial differential equations (PDEs)<n>We propose PIDO, a novel physics-informed neural PDE solver designed to generalize effectively across diverse PDE configurations.<n>We validate PIDO on a range of benchmarks, including 1D combined equations and 2D Navier-Stokes equations.
  arXiv  Detail & Related papers  (2024-11-28T13:16:20Z)
• Unisolver: PDE-Conditional Transformers Are Universal PDE Solvers [55.0876373185983]
  We present the Universal PDE solver (Unisolver) capable of solving a wide scope of PDEs. Our key finding is that a PDE solution is fundamentally under the control of a series of PDE components. Unisolver achieves consistent state-of-the-art results on three challenging large-scale benchmarks.
  arXiv  Detail & Related papers  (2024-05-27T15:34:35Z)
• Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEs [85.40198664108624]
  We propose Codomain Attention Neural Operator (CoDA-NO) to solve multiphysics problems with PDEs. CoDA-NO tokenizes functions along the codomain or channel space, enabling self-supervised learning or pretraining of multiple PDE systems. We find CoDA-NO to outperform existing methods by over 36% on complex downstream tasks with limited data.
  arXiv  Detail & Related papers  (2024-03-19T08:56:20Z)
• Deep Equilibrium Based Neural Operators for Steady-State PDEs [100.88355782126098]
  We study the benefits of weight-tied neural network architectures for steady-state PDEs. We propose FNO-DEQ, a deep equilibrium variant of the FNO architecture that directly solves for the solution of a steady-state PDE.
  arXiv  Detail & Related papers  (2023-11-30T22:34:57Z)
• Reduced-order modeling for parameterized PDEs via implicit neural representations [4.135710717238787]
  We present a new data-driven reduced-order modeling approach to efficiently solve parametrized partial differential equations (PDEs) The proposed framework encodes PDE and utilizes a parametrized neural ODE (PNODE) to learn latent dynamics characterized by multiple PDE parameters. We evaluate the proposed method at a large Reynolds number and obtain up to speedup of O(103) and 1% relative error to the ground truth values.
  arXiv  Detail & Related papers  (2023-11-28T01:35:06Z)
• 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)
• Learning time-dependent PDE solver using Message Passing Graph Neural Networks [0.0]
  We introduce a graph neural network approach to finding efficient PDE solvers through learning using message-passing models. We use graphs to represent PDE-data on an unstructured mesh and show that message passing graph neural networks (MPGNN) can parameterize governing equations. We show that a recurrent graph neural network approach can find a temporal sequence of solutions to a PDE.
  arXiv  Detail & Related papers  (2022-04-15T21:10:32Z)
• Neural Q-learning for solving PDEs [0.0]
  We develop a new numerical method for solving elliptic-type PDEs by adapting the Q-learning algorithm in reinforcement learning. Our "Q-PDE" algorithm is mesh-free and therefore has the potential to overcome the curse of dimensionality. The numerical performance of the Q-PDE algorithm is studied for several elliptic PDEs.
  arXiv  Detail & Related papers  (2022-03-31T15:52:44Z)
• NeuralPDE: Modelling Dynamical Systems from Data [0.44259821861543996]
  We propose NeuralPDE, a model which combines convolutional neural networks (CNNs) with differentiable ODE solvers to model dynamical systems. We show that the Method of Lines used in standard PDE solvers can be represented using convolutions which makes CNNs the natural choice to parametrize arbitrary PDE dynamics. Our model can be applied to any data without requiring any prior knowledge about the governing PDE.
  arXiv  Detail & Related papers  (2021-11-15T10:59:52Z)
• PDE-constrained Models with Neural Network Terms: Optimization and Global Convergence [0.0]
  Recent research has used deep learning to develop partial differential equation (PDE) models in science and engineering. We rigorously study the optimization of a class of linear elliptic PDEs with neural network terms. We train a neural network model for an application in fluid mechanics, in which the neural network functions as a closure model for the Reynolds-averaged Navier-Stokes equations.
  arXiv  Detail & Related papers  (2021-05-18T16:04:33Z)

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.