Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learning

• URL: http://arxiv.org/abs/2309.07946v1
• Date: Thu, 14 Sep 2023 14:10:22 GMT
• Title: Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learning
• Authors: Dimitrios G. Patsatzis, Gianluca Fabiani, Lucia Russo, Constantinos Siettos
• Abstract summary: We present a physics-informed machine-learning (PIML) approach for the approximation of slow invariant manifold (SIMs) of singularly perturbed systems. We show that the proposed PIML scheme provides approximations, of equivalent or even higher accuracy, than those provided by other traditional GSPT-based methods. A comparison of the computational costs between symbolic, automatic and numerical approximation of the required derivatives in the learning process is also provided.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present a physics-informed machine-learning (PIML) approach for the approximation of slow invariant manifolds (SIMs) of singularly perturbed systems, providing functionals in an explicit form that facilitate the construction and numerical integration of reduced order models (ROMs). The proposed scheme solves a partial differential equation corresponding to the invariance equation (IE) within the Geometric Singular Perturbation Theory (GSPT) framework. For the solution of the IE, we used two neural network structures, namely feedforward neural networks (FNNs), and random projection neural networks (RPNNs), with symbolic differentiation for the computation of the gradients required for the learning process. The efficiency of our PIML method is assessed via three benchmark problems, namely the Michaelis-Menten, the target mediated drug disposition reaction mechanism, and the 3D Sel'kov model. We show that the proposed PIML scheme provides approximations, of equivalent or even higher accuracy, than those provided by other traditional GSPT-based methods, and importantly, for any practical purposes, it is not affected by the magnitude of the perturbation parameter. This is of particular importance, as there are many systems for which the gap between the fast and slow timescales is not that big, but still ROMs can be constructed. A comparison of the computational costs between symbolic, automatic and numerical approximation of the required derivatives in the learning process is also provided.

Related papers

• Enhancing lattice kinetic schemes for fluid dynamics with Lattice-Equivariant Neural Networks [79.16635054977068]
  We present a new class of equivariant neural networks, dubbed Lattice-Equivariant Neural Networks (LENNs) Our approach develops within a recently introduced framework aimed at learning neural network-based surrogate models Lattice Boltzmann collision operators. Our work opens towards practical utilization of machine learning-augmented Lattice Boltzmann CFD in real-world simulations.
  arXiv  Detail & Related papers  (2024-05-22T17:23:15Z)
• A physics-informed neural network method for the approximation of slow invariant manifolds for the general class of stiff systems of ODEs [0.0]
  We present a physics-informed neural network (PINN) approach for the discovery of slow invariant manifold (SIMs) In contrast to other machine learning (ML) approaches that construct reduced order black box surrogate models, our approach, simultaneously decomposes the vector field into fast and slow components. We show that the proposed PINN scheme provides SIM approximations, of equivalent or even higher accuracy, than those provided by QSSA, PEA and CSP.
  arXiv  Detail & Related papers  (2024-03-18T09:10:39Z)
• Deep Learning-based surrogate models for parametrized PDEs: handling geometric variability through graph neural networks [0.0]
  This work explores the potential usage of graph neural networks (GNNs) for the simulation of time-dependent PDEs. We propose a systematic strategy to build surrogate models based on a data-driven time-stepping scheme. We show that GNNs can provide a valid alternative to traditional surrogate models in terms of computational efficiency and generalization to new scenarios.
  arXiv  Detail & Related papers  (2023-08-03T08:14:28Z)
• Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domains [0.0]
  Physics-informed neural networks (PINNs) are a new tool for solving boundary value problems. Recent investigations have shown that when designing loss functions for many engineering problems, using first-order derivatives and combining equations from both strong and weak forms can lead to much better accuracy. In this work, we propose applying the mixed formulation to solve multi-physical problems, specifically a stationary thermo-mechanically coupled system of equations.
  arXiv  Detail & Related papers  (2023-02-09T21:56:59Z)
• Deep learning applied to computational mechanics: A comprehensive review, state of the art, and the classics [77.34726150561087]
  Recent developments in artificial neural networks, particularly deep learning (DL), are reviewed in detail. Both hybrid and pure machine learning (ML) methods are discussed. History and limitations of AI are recounted and discussed, with particular attention at pointing out misstatements or misconceptions of the classics.
  arXiv  Detail & Related papers  (2022-12-18T02:03:00Z)
• Accelerated Solutions of Coupled Phase-Field Problems using Generative Adversarial Networks [0.0]
  We develop a new neural network based framework that uses encoder-decoder based conditional GeneLSTM layers to solve a system of Cahn-Hilliard microstructural equations. We show that the trained models are mesh and scale-independent, thereby warranting application as effective neural operators.
  arXiv  Detail & Related papers  (2022-11-22T08:32:22Z)
• A deep learning driven pseudospectral PCE based FFT homogenization algorithm for complex microstructures [68.8204255655161]
  It is shown that the proposed method is able to predict central moments of interest while being magnitudes faster to evaluate than traditional approaches. It is shown, that the proposed method is able to predict central moments of interest while being magnitudes faster to evaluate than traditional approaches.
  arXiv  Detail & Related papers  (2021-10-26T07:02:14Z)
• 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)
• Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach [144.21892195917758]
  We study estimation in a class of generalized Structural equation models (SEMs) We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using a gradient descent. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.
  arXiv  Detail & Related papers  (2020-07-02T17:55:47Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)

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.