UniFIDES: Universal Fractional Integro-Differential Equation Solvers

• URL: http://arxiv.org/abs/2407.01848v2
• Date: Mon, 8 Jul 2024 13:18:17 GMT
• Title: UniFIDES: Universal Fractional Integro-Differential Equation Solvers
• Authors: Milad Saadat, Deepak Mangal, Safa Jamali,
• Abstract summary: This work introduces the Universal Fractional Integro-Differential Equation Solvers (UniFIDES) UniFIDES is a comprehensive machine learning platform designed to expeditiously solve a variety of FIDEs in both forward and inverse directions. Our results highlight UniFIDES' ability to accurately solve a wide spectrum of integro-differential equations and offer the prospect of using machine learning platforms universally.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The development of data-driven approaches for solving differential equations has been followed by a plethora of applications in science and engineering across a multitude of disciplines and remains a central focus of active scientific inquiry. However, a large body of natural phenomena incorporates memory effects that are best described via fractional integro-differential equations (FIDEs), in which the integral or differential operators accept non-integer orders. Addressing the challenges posed by nonlinear FIDEs is a recognized difficulty, necessitating the application of generic methods with immediate practical relevance. This work introduces the Universal Fractional Integro-Differential Equation Solvers (UniFIDES), a comprehensive machine learning platform designed to expeditiously solve a variety of FIDEs in both forward and inverse directions, without the need for ad hoc manipulation of the equations. The effectiveness of UniFIDES is demonstrated through a collection of integer-order and fractional problems in science and engineering. Our results highlight UniFIDES' ability to accurately solve a wide spectrum of integro-differential equations and offer the prospect of using machine learning platforms universally for discovering and describing dynamical and complex systems.

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)
• Transfer Operator Learning with Fusion Frame [0.0]
  This work presents a novel framework that enhances the transfer learning capabilities of operator learning models for solving Partial Differential Equations (PDEs) We introduce an innovative architecture that combines fusion frames with POD-DeepONet, demonstrating superior performance across various PDEs in our experimental analysis. Our framework addresses the critical challenge of transfer learning in operator learning models, paving the way for adaptable and efficient solutions across a wide range of scientific and engineering applications.
  arXiv  Detail & Related papers  (2024-08-20T00:03:23Z)
• Assessment of Uncertainty Quantification in Universal Differential Equations [1.374796982212312]
  Universal Differential Equations (UDEs) are used to combine prior knowledge in the form of mechanistic formulations with universal function approximators, like neural networks. We provide a formalisation of uncertainty quantification (UQ) for UDEs and investigate important frequentist and Bayesian methods.
  arXiv  Detail & Related papers  (2024-06-13T06:36:19Z)
• Towards true discovery of the differential equations [57.089645396998506]
  Differential equation discovery is a machine learning subfield used to develop interpretable models. This paper explores the prerequisites and tools for independent equation discovery without expert input.
  arXiv  Detail & Related papers  (2023-08-09T12:03:12Z)
• AttNS: Attention-Inspired Numerical Solving For Limited Data Scenarios [51.94807626839365]
  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.
  arXiv  Detail & Related papers  (2023-02-05T01:39:21Z)
• Symbolic Recovery of Differential Equations: The Identifiability Problem [52.158782751264205]
  Symbolic recovery of differential equations is the ambitious attempt at automating the derivation of governing equations. We provide both necessary and sufficient conditions for a function to uniquely determine the corresponding differential equation. We then use our results to devise numerical algorithms aiming to determine whether a function solves a differential equation uniquely.
  arXiv  Detail & Related papers  (2022-10-15T17:32:49Z)
• 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)
• Neural Integral Equations [3.087238735145305]
  We introduce a method for learning unknown integral operators from data using an IE solver. We also present Attentional Neural Integral Equations (ANIE), which replaces the integral with self-attention.
  arXiv  Detail & Related papers  (2022-09-30T02:32:17Z)
• D-CIPHER: Discovery of Closed-form Partial Differential Equations [80.46395274587098]
  We propose D-CIPHER, which is robust to measurement artifacts and can uncover a new and very general class of differential equations. We further design a novel optimization procedure, CoLLie, to help D-CIPHER search through this class efficiently.
  arXiv  Detail & Related papers  (2022-06-21T17:59:20Z)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)

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.