AttNS: Attention-Inspired Numerical Solving For Limited Data Scenarios

• URL: http://arxiv.org/abs/2302.10184v2
• Date: Wed, 05 Feb 2025 12:23:15 GMT
• Title: AttNS: Attention-Inspired Numerical Solving For Limited Data Scenarios
• Authors: Zhongzhan Huang, Mingfu Liang, Shanshan Zhong, Liang Lin,
• Abstract summary: We propose the attention-inspired numerical solver (AttNS) to solve differential equations due to limited data. AttNS is inspired by the effectiveness of attention modules in Residual Neural Networks (ResNet) in enhancing model generalization and robustness.
• Score: 51.94807626839365
• License:
• Abstract: We propose the attention-inspired numerical solver (AttNS), a concise method that helps the generalization and robustness issues faced by the AI-Hybrid numerical solver in solving differential equations due to limited data. AttNS is inspired by the effectiveness of attention modules in Residual Neural Networks (ResNet) in enhancing model generalization and robustness for conventional deep learning tasks. Drawing from the dynamical system perspective of ResNet, we seamlessly incorporate attention mechanisms into the design of numerical methods tailored for the characteristics of solving differential equations. Our results on benchmarks, ranging from high-dimensional problems to chaotic systems, showcases AttNS consistently enhancing various numerical solvers without any intricate model crafting. Finally, we analyze AttNS experimentally and theoretically, demonstrating its ability to achieve strong generalization and robustness while ensuring the convergence of the solver. This includes requiring less data compared to other advanced methods to achieve comparable generalization errors and better prevention of numerical explosion issues when solving differential equations.

Related papers

• Advanced Physics-Informed Neural Network with Residuals for Solving Complex Integral Equations [0.13499500088995461]
  RISN is a novel neural network architecture designed to solve a wide range of integral and integro-differential equations. We show that RISN consistently outperforms PINN, achieving significantly lower Mean Absolute Errors (MAE) across various types of equations. The results highlight RISN's robustness and efficiency in solving challenging integral and integro-differential problems.
  arXiv  Detail & Related papers  (2025-01-22T19:47:03Z)
• A Model-Constrained Discontinuous Galerkin Network (DGNet) for Compressible Euler Equations with Out-of-Distribution Generalization [0.0]
  We develop a model-constrained discontinuous Galerkin Network (DGNet) approach. The core of DGNet is the synergy of several key strategies. We present comprehensive numerical results for 1D and 2D compressible Euler equation problems.
  arXiv  Detail & Related papers  (2024-09-27T01:13:38Z)
• Physics Informed Kolmogorov-Arnold Neural Networks for Dynamical Analysis via Efficent-KAN and WAV-KAN [0.12045539806824918]
  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.
  arXiv  Detail & Related papers  (2024-07-25T20:14:58Z)
• 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)
• Machine Learning Insides OptVerse AI Solver: Design Principles and Applications [74.67495900436728]
  We present a comprehensive study on the integration of machine learning (ML) techniques into Huawei Cloud's OptVerse AI solver. We showcase our methods for generating complex SAT and MILP instances utilizing generative models that mirror multifaceted structures of real-world problem. We detail the incorporation of state-of-the-art parameter tuning algorithms which markedly elevate solver performance.
  arXiv  Detail & Related papers  (2024-01-11T15:02:15Z)
• Enhancing Low-Order Discontinuous Galerkin Methods with Neural Ordinary Differential Equations for Compressible Navier--Stokes Equations [0.1578515540930834]
  We introduce an end-to-end differentiable framework for solving the compressible Navier-Stokes equations. This integrated approach combines a differentiable discontinuous Galerkin solver with a neural network source term. We demonstrate the performance of the proposed framework through two examples.
  arXiv  Detail & Related papers  (2023-10-29T04:26:23Z)
• Physical Information Neural Networks for Solving High-index Differential-algebraic Equation Systems Based on Radau Methods [10.974537885042613]
  We propose a PINN computational framework, combined Radau IIA numerical method with a neural network structure via the attention mechanisms, to directly solve high-index DAEs. Our method exhibits excellent computational accuracy and strong generalization capabilities, providing a feasible approach for the high-precision solution of larger-scale DAEs.
  arXiv  Detail & Related papers  (2023-10-19T15:57:10Z)
• 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)
• 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)
• Generalization of Neural Combinatorial Solvers Through the Lens of Adversarial Robustness [68.97830259849086]
  Most datasets only capture a simpler subproblem and likely suffer from spurious features. We study adversarial robustness - a local generalization property - to reveal hard, model-specific instances and spurious features. Unlike in other applications, where perturbation models are designed around subjective notions of imperceptibility, our perturbation models are efficient and sound. Surprisingly, with such perturbations, a sufficiently expressive neural solver does not suffer from the limitations of the accuracy-robustness trade-off common in supervised learning.
  arXiv  Detail & Related papers  (2021-10-21T07:28:11Z)

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.