Partial Differential Equations is All You Need for Generating Neural Architectures -- A Theory for Physical Artificial Intelligence Systems

• URL: http://arxiv.org/abs/2103.08313v1
• Date: Wed, 10 Mar 2021 00:05:46 GMT
• Title: Partial Differential Equations is All You Need for Generating Neural Architectures -- A Theory for Physical Artificial Intelligence Systems
• Authors: Ping Guo, Kaizhu Huang, and Zenglin Xu
• Abstract summary: We generalize the reaction-diffusion equation in statistical physics, Schr"odinger equation in quantum mechanics, Helmholtz equation in paraxial optics. We take finite difference method to discretize NPDE for finding numerical solution. Basic building blocks of deep neural network architecture, including multi-layer perceptron, convolutional neural network and recurrent neural networks, are generated.
• Score: 29.667065357274385
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: In this work, we generalize the reaction-diffusion equation in statistical physics, Schr\"odinger equation in quantum mechanics, Helmholtz equation in paraxial optics into the neural partial differential equations (NPDE), which can be considered as the fundamental equations in the field of artificial intelligence research. We take finite difference method to discretize NPDE for finding numerical solution, and the basic building blocks of deep neural network architecture, including multi-layer perceptron, convolutional neural network and recurrent neural networks, are generated. The learning strategies, such as Adaptive moment estimation, L-BFGS, pseudoinverse learning algorithms and partial differential equation constrained optimization, are also presented. We believe it is of significance that presented clear physical image of interpretable deep neural networks, which makes it be possible for applying to analog computing device design, and pave the road to physical artificial intelligence.

Related papers

• Chebyshev Spectral Neural Networks for Solving Partial Differential Equations [0.0]
  The study uses a feedforward neural network model and error backpropagation principles, utilizing automatic differentiation (AD) to compute the loss function. The numerical efficiency and accuracy of the CSNN model are investigated through testing on elliptic partial differential equations, and it is compared with the well-known Physics-Informed Neural Network(PINN) method.
  arXiv  Detail & Related papers  (2024-06-06T05:31:45Z)
• Graph Neural Networks for Learning Equivariant Representations of Neural Networks [55.04145324152541]
  We propose to represent neural networks as computational graphs of parameters. Our approach enables a single model to encode neural computational graphs with diverse architectures. We showcase the effectiveness of our method on a wide range of tasks, including classification and editing of implicit neural representations.
  arXiv  Detail & Related papers  (2024-03-18T18:01:01Z)
• Mechanistic Neural Networks for Scientific Machine Learning [58.99592521721158]
  We present Mechanistic Neural Networks, a neural network design for machine learning applications in the sciences. It incorporates a new Mechanistic Block in standard architectures to explicitly learn governing differential equations as representations. Central to our approach is a novel Relaxed Linear Programming solver (NeuRLP) inspired by a technique that reduces solving linear ODEs to solving linear programs.
  arXiv  Detail & Related papers  (2024-02-20T15:23:24Z)
• PMNN:Physical Model-driven Neural Network for solving time-fractional differential equations [17.66402435033991]
  An innovative Physical Model-driven Neural Network (PMNN) method is proposed to solve time-fractional differential equations. It effectively combines deep neural networks (DNNs) with approximation of fractional derivatives.
  arXiv  Detail & Related papers  (2023-10-07T12:43:32Z)
• Spectral-Bias and Kernel-Task Alignment in Physically Informed Neural Networks [4.604003661048267]
  Physically informed neural networks (PINNs) are a promising emerging method for solving differential equations. We propose a comprehensive theoretical framework that sheds light on this important problem. We derive an integro-differential equation that governs PINN prediction in the large data-set limit.
  arXiv  Detail & Related papers  (2023-07-12T18:00:02Z)
• NeuralStagger: Accelerating Physics-constrained Neural PDE Solver with Spatial-temporal Decomposition [67.46012350241969]
  This paper proposes a general acceleration methodology called NeuralStagger. It decomposing the original learning tasks into several coarser-resolution subtasks. We demonstrate the successful application of NeuralStagger on 2D and 3D fluid dynamics simulations.
  arXiv  Detail & Related papers  (2023-02-20T19:36:52Z)
• Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations [64.78260098263489]
  In this work, we assess the ability of physics-informed neural networks (PINNs) to solve increasingly-complex coupled ordinary differential equations (ODEs) We show that PINNs eventually fail to produce correct solutions to these benchmarks as their complexity increases. We identify several reasons why this may be the case, including insufficient network capacity, poor conditioning of the ODEs, and high local curvature, as measured by the Laplacian of the PINN loss.
  arXiv  Detail & Related papers  (2022-10-14T15:01:32Z)
• Unsupervised Legendre-Galerkin Neural Network for Stiff Partial Differential Equations [9.659504024299896]
  We propose an unsupervised machine learning algorithm based on the Legendre-Galerkin neural network to find an accurate approximation to the solution of different types of PDEs. The proposed neural network is applied to the general 1D and 2D PDEs as well as singularly perturbed PDEs that possess boundary layer behavior.
  arXiv  Detail & Related papers  (2022-07-21T00:47:47Z)
• Physics Informed RNN-DCT Networks for Time-Dependent Partial Differential Equations [62.81701992551728]
  We present a physics-informed framework for solving time-dependent partial differential equations. Our model utilizes discrete cosine transforms to encode spatial and recurrent neural networks. We show experimental results on the Taylor-Green vortex solution to the Navier-Stokes equations.
  arXiv  Detail & Related papers  (2022-02-24T20:46:52Z)
• A deep learning theory for neural networks grounded in physics [2.132096006921048]
  We argue that building large, fast and efficient neural networks on neuromorphic architectures requires rethinking the algorithms to implement and train them. Our framework applies to a very broad class of models, namely systems whose state or dynamics are described by variational equations.
  arXiv  Detail & Related papers  (2021-03-18T02:12:48Z)
• 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.