Evidential Physics-Informed Neural Networks

• URL: http://arxiv.org/abs/2501.15908v1
• Date: Mon, 27 Jan 2025 10:01:10 GMT
• Title: Evidential Physics-Informed Neural Networks
• Authors: Hai Siong Tan, Kuancheng Wang, Rafe McBeth,
• Abstract summary: We present a novel class of Physics-Informed Neural Networks that is formulated based on the principles of Evidential Deep Learning. We show how to apply our model to inverse problems involving 1D and 2D nonlinear differential equations.
• Score: 0.0
• License:
• Abstract: We present a novel class of Physics-Informed Neural Networks that is formulated based on the principles of Evidential Deep Learning, where the model incorporates uncertainty quantification by learning parameters of a higher-order distribution. The dependent and trainable variables of the PDE residual loss and data-fitting loss terms are recast as functions of the hyperparameters of an evidential prior distribution. Our model is equipped with an information-theoretic regularizer that contains the Kullback-Leibler divergence between two inverse-gamma distributions characterizing predictive uncertainty. Relative to Bayesian-Physics-Informed-Neural-Networks, our framework appeared to exhibit higher sensitivity to data noise, preserve boundary conditions more faithfully and yield empirical coverage probabilities closer to nominal ones. Toward examining its relevance for data mining in scientific discoveries, we demonstrate how to apply our model to inverse problems involving 1D and 2D nonlinear differential equations.

Related papers

• Physics-Informed Neural Network based inverse framework for time-fractional differential equations for rheology [0.0]
  Time-fractional differential equations offer a robust framework for capturing phenomena characterized by memory effects. However, solving inverse problems involving fractional derivatives presents notable challenges, including issues related to stability and uniqueness. In this study, we extend the application of PINNs to address inverse problems involving time-fractional derivatives, specifically targeting two problems: 1) anomalous diffusion and 2) fractional viscoelastic equation.
  arXiv  Detail & Related papers  (2024-06-06T01:29:17Z)
• DiffHybrid-UQ: Uncertainty Quantification for Differentiable Hybrid Neural Modeling [4.76185521514135]
  We introduce a novel method, DiffHybrid-UQ, for effective and efficient uncertainty propagation and estimation in hybrid neural differentiable models. Specifically, our approach effectively discerns and quantifies both aleatoric uncertainties, arising from data noise, and epistemic uncertainties, resulting from model-form discrepancies and data sparsity.
  arXiv  Detail & Related papers  (2023-12-30T07:40:47Z)
• Assessing Neural Network Representations During Training Using Noise-Resilient Diffusion Spectral Entropy [55.014926694758195]
  Entropy and mutual information in neural networks provide rich information on the learning process. We leverage data geometry to access the underlying manifold and reliably compute these information-theoretic measures. We show that they form noise-resistant measures of intrinsic dimensionality and relationship strength in high-dimensional simulated data.
  arXiv  Detail & Related papers  (2023-12-04T01:32:42Z)
• SimPINNs: Simulation-Driven Physics-Informed Neural Networks for Enhanced Performance in Nonlinear Inverse Problems [0.0]
  This paper introduces a novel approach to solve inverse problems by leveraging deep learning techniques. The objective is to infer unknown parameters that govern a physical system based on observed data.
  arXiv  Detail & Related papers  (2023-09-27T06:34:55Z)
• Efficient Bayesian inference using physics-informed invertible neural networks for inverse problems [6.97393424359704]
  We introduce an innovative approach for addressing Bayesian inverse problems through the utilization of physics-informed invertible neural networks (PI-INN) The PI-INN offers a precise and efficient generative model for Bayesian inverse problems, yielding tractable posterior density estimates. As a particular physics-informed deep learning model, the primary training challenge for PI-INN centers on enforcing the independence constraint.
  arXiv  Detail & Related papers  (2023-04-25T03:17:54Z)
• Capturing dynamical correlations using implicit neural representations [85.66456606776552]
  We develop an artificial intelligence framework which combines a neural network trained to mimic simulated data from a model Hamiltonian with automatic differentiation to recover unknown parameters from experimental data. In doing so, we illustrate the ability to build and train a differentiable model only once, which then can be applied in real-time to multi-dimensional scattering data.
  arXiv  Detail & Related papers  (2023-04-08T07:55:36Z)
• Evaluating the Adversarial Robustness for Fourier Neural Operators [78.36413169647408]
  Fourier Neural Operator (FNO) was the first to simulate turbulent flow with zero-shot super-resolution. We generate adversarial examples for FNO based on norm-bounded data input perturbations. Our results show that the model's robustness degrades rapidly with increasing perturbation levels.
  arXiv  Detail & Related papers  (2022-04-08T19:19:42Z)
• Mixed Effects Neural ODE: A Variational Approximation for Analyzing the Dynamics of Panel Data [50.23363975709122]
  We propose a probabilistic model called ME-NODE to incorporate (fixed + random) mixed effects for analyzing panel data. We show that our model can be derived using smooth approximations of SDEs provided by the Wong-Zakai theorem. We then derive Evidence Based Lower Bounds for ME-NODE, and develop (efficient) training algorithms.
  arXiv  Detail & Related papers  (2022-02-18T22:41:51Z)
• Learning Physics-Informed Neural Networks without Stacked Back-propagation [82.26566759276105]
  We develop a novel approach that can significantly accelerate the training of Physics-Informed Neural Networks. In particular, we parameterize the PDE solution by the Gaussian smoothed model and show that, derived from Stein's Identity, the second-order derivatives can be efficiently calculated without back-propagation. Experimental results show that our proposed method can achieve competitive error compared to standard PINN training but is two orders of magnitude faster.
  arXiv  Detail & Related papers  (2022-02-18T18:07:54Z)
• The Interplay Between Implicit Bias and Benign Overfitting in Two-Layer Linear Networks [51.1848572349154]
  neural network models that perfectly fit noisy data can generalize well to unseen test data. We consider interpolating two-layer linear neural networks trained with gradient flow on the squared loss and derive bounds on the excess risk.
  arXiv  Detail & Related papers  (2021-08-25T22:01:01Z)
• Error Bounds of the Invariant Statistics in Machine Learning of Ergodic It\^o Diffusions [8.627408356707525]
  We study the theoretical underpinnings of machine learning of ergodic Ito diffusions. We deduce a linear dependence of the errors of one-point and two-point invariant statistics on the error in the learning of the drift and diffusion coefficients.
  arXiv  Detail & Related papers  (2021-05-21T02:55:59Z)

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.