Kernel-Adaptive PI-ELMs for Forward and Inverse Problems in PDEs with Sharp Gradients

• URL: http://arxiv.org/abs/2507.10241v1
• Date: Mon, 14 Jul 2025 13:03:53 GMT
• Title: Kernel-Adaptive PI-ELMs for Forward and Inverse Problems in PDEs with Sharp Gradients
• Authors: Vikas Dwivedi, Balaji Srinivasan, Monica Sigovan, Bruno Sixou,
• Abstract summary: This paper introduces the Kernel Adaptive Physics-Informed Extreme Learning Machine (KAPI-ELM)<n>It is designed to solve both forward and inverse Partial Differential Equation (PDE) problems involving localized sharp gradients.<n>KAPI-ELM achieves state-of-the-art accuracy in both forward and inverse settings.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This paper introduces the Kernel Adaptive Physics-Informed Extreme Learning Machine (KAPI-ELM), an adaptive Radial Basis Function (RBF)-based extension of PI-ELM designed to solve both forward and inverse Partial Differential Equation (PDE) problems involving localized sharp gradients. While PI-ELMs outperform the traditional Physics-Informed Neural Networks (PINNs) in speed due to their single-shot, least square optimization, this advantage comes at a cost: their fixed, randomly initialized input layer limits their ability to capture sharp gradients. To overcome this limitation, we introduce a lightweight Bayesian Optimization (BO) framework that, instead of adjusting each input layer parameter individually as in traditional backpropagation, learns a small set of hyperparameters defining the statistical distribution from which the input weights are drawn. This novel distributional optimization strategy -- combining BO for input layer distributional parameters with least-squares optimization for output layer network parameters -- enables KAPI-ELM to preserve PI-ELM's speed while matching or exceeding the expressiveness of PINNs. We validate the proposed methodology on several challenging forward and inverse PDE benchmarks, including a 1D singularly perturbed convection-diffusion equation, a 2D Poisson equation with sharp localized sources, and a time-dependent advection equation. Notably, KAPI-ELM achieves state-of-the-art accuracy in both forward and inverse settings. In stiff PDE regimes, it matches or even outperforms advanced methods such as the Extended Theory of Functional Connections (XTFC), while requiring nearly an order of magnitude fewer tunable parameters. These results establish the potential of KAPI-ELM as a scalable, interpretable, and generalizable physics-informed learning framework, especially in stiff PDE regimes.

Related papers

• PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations [5.4087282763977855]
  We propose Physics-Informed Gaussians (PIGs), which combine feature embeddings using Gaussian functions with a lightweight neural network.<n>Our approach uses trainable parameters for the mean and variance of each Gaussian, allowing for dynamic adjustment of their positions and shapes during training.<n> Experimental results show the competitive performance of our model across various PDEs, demonstrating its potential as a robust tool for solving complex PDEs.
  arXiv  Detail & Related papers  (2024-12-08T16:58:29Z)
• Local Loss Optimization in the Infinite Width: Stable Parameterization of Predictive Coding Networks and Target Propagation [8.35644084613785]
  We introduce the maximal update parameterization ($mu$P) in the infinite-width limit for two representative designs of local targets.<n>By analyzing deep linear networks, we found that PC's gradients interpolate between first-order and Gauss-Newton-like gradients.<n>We demonstrate that, in specific standard settings, PC in the infinite-width limit behaves more similarly to the first-order gradient.
  arXiv  Detail & Related papers  (2024-11-04T11:38:27Z)
• Physics informed cell representations for variational formulation of multiscale problems [8.905008042172883]
  Physics-Informed Neural Networks (PINNs) are emerging as a promising tool for solving partial differential equations (PDEs) PINNs are not well suited for solving PDEs with multiscale features, particularly suffering from slow convergence and poor accuracy. This article proposes a cell-based model architecture consisting of multilevel multiresolution grids coupled with a multilayer perceptron (MLP) In essence, by cell-based model along with the parallel tiny-cuda-nn library, our implementation improves convergence speed and numerical accuracy.
  arXiv  Detail & Related papers  (2024-05-27T02:42:16Z)
• Efficiently Training Deep-Learning Parametric Policies using Lagrangian Duality [55.06411438416805]
  Constrained Markov Decision Processes (CMDPs) are critical in many high-stakes applications.<n>This paper introduces a novel approach, Two-Stage Deep Decision Rules (TS- DDR) to efficiently train parametric actor policies.<n>It is shown to enhance solution quality and to reduce computation times by several orders of magnitude when compared to current state-of-the-art methods.
  arXiv  Detail & Related papers  (2024-05-23T18:19:47Z)
• RoPINN: Region Optimized Physics-Informed Neural Networks [66.38369833561039]
  Physics-informed neural networks (PINNs) have been widely applied to solve partial differential equations (PDEs) This paper proposes and theoretically studies a new training paradigm as region optimization. A practical training algorithm, Region Optimized PINN (RoPINN), is seamlessly derived from this new paradigm.
  arXiv  Detail & Related papers  (2024-05-23T09:45:57Z)
• Differentially Private Optimization with Sparse Gradients [60.853074897282625]
  We study differentially private (DP) optimization problems under sparsity of individual gradients. Building on this, we obtain pure- and approximate-DP algorithms with almost optimal rates for convex optimization with sparse gradients.
  arXiv  Detail & Related papers  (2024-04-16T20:01:10Z)
• An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensions [1.2981626828414923]
  We present two effective methods for solving high-dimensional partial differential equations (PDE) based on randomized neural networks. We present ample numerical simulations for a number of high-dimensional linear/nonlinear stationary/dynamic PDEs to demonstrate their performance.
  arXiv  Detail & Related papers  (2023-09-13T15:59:02Z)
• Neural Basis Functions for Accelerating Solutions to High Mach Euler Equations [63.8376359764052]
  We propose an approach to solving partial differential equations (PDEs) using a set of neural networks. We regress a set of neural networks onto a reduced order Proper Orthogonal Decomposition (POD) basis. These networks are then used in combination with a branch network that ingests the parameters of the prescribed PDE to compute a reduced order approximation to the PDE.
  arXiv  Detail & Related papers  (2022-08-02T18:27:13Z)
• 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)
• Physics-Informed Neural Operator for Learning Partial Differential Equations [55.406540167010014]
  PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families.
  arXiv  Detail & Related papers  (2021-11-06T03:41:34Z)
• Adaptive Subcarrier, Parameter, and Power Allocation for Partitioned Edge Learning Over Broadband Channels [69.18343801164741]
  partitioned edge learning (PARTEL) implements parameter-server training, a well known distributed learning method, in wireless network. We consider the case of deep neural network (DNN) models which can be trained using PARTEL by introducing some auxiliary variables.
  arXiv  Detail & Related papers  (2020-10-08T15:27:50Z)

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.