Physics Informed Kolmogorov-Arnold Neural Networks for Dynamical Analysis via Efficent-KAN and WAV-KAN

• URL: http://arxiv.org/abs/2407.18373v2
• Date: Mon, 29 Jul 2024 03:46:15 GMT
• Title: Physics Informed Kolmogorov-Arnold Neural Networks for Dynamical Analysis via Efficent-KAN and WAV-KAN
• Authors: Subhajit Patra, Sonali Panda, Bikram Keshari Parida, Mahima Arya, Kurt Jacobs, Denys I. Bondar, Abhijit Sen,
• Abstract summary: We implement the Physics-Informed Kolmogorov-Arnold Neural Networks (PIKAN) through efficient-KAN and WAV-KAN. PIKAN demonstrates superior performance compared to conventional deep neural networks, achieving the same level of accuracy with fewer layers and reduced computational overhead.
• Score: 0.12045539806824918
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Physics-informed neural networks have proven to be a powerful tool for solving differential equations, leveraging the principles of physics to inform the learning process. However, traditional deep neural networks often face challenges in achieving high accuracy without incurring significant computational costs. In this work, we implement the Physics-Informed Kolmogorov-Arnold Neural Networks (PIKAN) through efficient-KAN and WAV-KAN, which utilize the Kolmogorov-Arnold representation theorem. PIKAN demonstrates superior performance compared to conventional deep neural networks, achieving the same level of accuracy with fewer layers and reduced computational overhead. We explore both B-spline and wavelet-based implementations of PIKAN and benchmark their performance across various ordinary and partial differential equations using unsupervised (data-free) and supervised (data-driven) techniques. For certain differential equations, the data-free approach suffices to find accurate solutions, while in more complex scenarios, the data-driven method enhances the PIKAN's ability to converge to the correct solution. We validate our results against numerical solutions and achieve $99 \%$ accuracy in most scenarios.

Related papers

• DiffGrad for Physics-Informed Neural Networks [0.0]
  Burgers' equation, a fundamental equation in fluid dynamics that is extensively used in PINNs, provides flexible results with the Adamprop. This paper introduces a novel strategy for solving Burgers' equation by incorporating DiffGrad with PINNs.
  arXiv  Detail & Related papers  (2024-09-05T04:39:35Z)
• 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)
• Knowledge-Based Convolutional Neural Network for the Simulation and Prediction of Two-Phase Darcy Flows [3.5707423185282656]
  Physics-informed neural networks (PINNs) have gained significant prominence as a powerful tool in the field of scientific computing and simulations. We propose to combine the power of neural networks with the dynamics imposed by the discretized differential equations. By discretizing the governing equations, the PINN learns to account for the discontinuities and accurately capture the underlying relationships between inputs and outputs.
  arXiv  Detail & Related papers  (2024-04-04T06:56:32Z)
• 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)
• Implicit Stochastic Gradient Descent for Training Physics-informed Neural Networks [51.92362217307946]
  Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems. PINNs are trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features. In this paper, we propose to employ implicit gradient descent (ISGD) method to train PINNs for improving the stability of training process.
  arXiv  Detail & Related papers  (2023-03-03T08:17:47Z)
• 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)
• Physics-aware deep learning framework for linear elasticity [0.0]
  The paper presents an efficient and robust data-driven deep learning (DL) computational framework for linear continuum elasticity problems. For an accurate representation of the field variables, a multi-objective loss function is proposed. Several benchmark problems including the Airimaty solution to elasticity and the Kirchhoff-Love plate problem are solved.
  arXiv  Detail & Related papers  (2023-02-19T20:33:32Z)
• On Robust Numerical Solver for ODE via Self-Attention Mechanism [82.95493796476767]
  We explore training efficient and robust AI-enhanced numerical solvers with a small data size by mitigating intrinsic noise disturbances. We first analyze the ability of the self-attention mechanism to regulate noise in supervised learning and then propose a simple-yet-effective numerical solver, Attr, which introduces an additive self-attention mechanism to the numerical solution of differential equations.
  arXiv  Detail & Related papers  (2023-02-05T01:39:21Z)
• Transfer Learning with Physics-Informed Neural Networks for Efficient Simulation of Branched Flows [1.1470070927586016]
  Physics-Informed Neural Networks (PINNs) offer a promising approach to solving differential equations. We adopt a recently developed transfer learning approach for PINNs and introduce a multi-head model. We show that our methods provide significant computational speedups in comparison to standard PINNs trained from scratch.
  arXiv  Detail & Related papers  (2022-11-01T01:50:00Z)
• 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)
• Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction [79.81193813215872]
  We develop a hybrid (graph) neural network that combines a traditional graph convolutional network with an embedded differentiable fluid dynamics simulator inside the network itself. We show that we can both generalize well to new situations and benefit from the substantial speedup of neural network CFD predictions.
  arXiv  Detail & Related papers  (2020-07-08T21:23:19Z)

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.