Kapil Chawla

Kapil Chawla, Youngjoon Hong, Jae Yong Lee et al.

We introduce Discontinuous Galerkin Finite Element Operator Network (DG--FEONet), a data-free operator learning framework that combines the strengths of the discontinuous Galerkin (DG) method with neural networks to solve parametric partial differential equations (PDEs) with discontinuous coefficients and non-smooth solutions. Unlike traditional operator learning models such as DeepONet and Fourier Neural Operator, which require large paired datasets and often struggle near sharp features, our approach minimizes the residual of a DG-based weak formulation using the Symmetric Interior Penalty Galerkin (SIPG) scheme. DG-FEONet predicts element-wise solution coefficients via a neural network, enabling data-free training without the need for precomputed input-output pairs. We provide theoretical justification through convergence analysis and validate the model's performance on a series of one- and two-dimensional PDE problems, demonstrating accurate recovery of discontinuities, strong generalization across parameter space, and reliable convergence rates. Our results highlight the potential of combining local discretization schemes with machine learning to achieve robust, singularity-aware operator approximation in challenging PDE settings.

Kapil Chawla, Sanghyun Lee, Yeonjong Shin

Material properties such as permeability fields in heterogeneous porous media are often represented as discontinuous, piecewise constant data tied to a given spatial discretization. Such representations are inherently mesh-dependent, requiring interpolation or projection whenever they are transferred to a different discretization. In this work, we develop \emph{Parallel and Adaptive Mesh-Free Approximation (PAM)}, a mesh-independent framework that approximates discontinuous data by a continuous, closed-form function. The resulting approximation can be evaluated consistently across different geometries and numerical discretizations, while preserving sharp interface features. The proposed PAM framework employs radial basis functions (RBFs) to construct continuous approximations of discontinuous data. To accurately capture discontinuities, we incorporate Shepard-normalization, which stabilizes the approximation near sharp interfaces. The coefficients of the RBF expansion are determined via sparse regression, enabling automatic selection of the most relevant basis functions and promoting robust representations. In addition, we develop a novel adaptive refinement approach which further enriches the approximation in regions of rapid spatial variation. We provide a theoretical analysis showing that the proposed normalized RBF framework achieves arbitrarily small $L^1$ error in approximating discontinuous step functions. To enhance computational efficiency, the domain is partitioned into subdomains, and the reconstruction problem is solved independently on each subdomain in parallel. Numerical experiments demonstrate the accuracy, adaptivity, and scalability of the proposed method, including applications to challenging heterogeneous permeability fields.

Kapil Chawla, William Holmes

In this article we develop a Physics Informed Neural Network (PINN) approach to simulate ice sheet dynamics governed by the Shallow Ice Approximation. This problem takes the form of a time-dependent parabolic obstacle problem. Prior work has used this approach to address the stationary obstacle problem and here we extend it to the time dependent problem. Through comprehensive 1D and 2D simulations, we validate the model's effectiveness in capturing complex free-boundary conditions. By merging traditional mathematical modeling with cutting-edge deep learning methods, this approach provides a scalable and robust solution for predicting temporal variations in ice thickness. To illustrate this approach in a real world setting, we simulate the dynamics of the Devon Ice Cap, incorporating aerogeophysical data from 2000 and 2018.