Tomasz Służalec

Maciej Paszyński, Tomasz Służalec

Physics-informed neural networks (PINNs) formulate the solution of partial differential equations as residual minimization problems over neural network parameterizations. Although highly flexible, optimization of PINNs using modern variants of Stochastic Gradient Descent algorithms is expensive. On the other hand, iterative computation of PINN parameterization using the Gauss-Newton method suffers from convergence difficulties, dense Jacobian structures, and poor conditioning that limit the effectiveness of second-order optimization methods. In this work, we introduce IGA-ODIL, a spline-based residual minimization framework combining ideas from Optimizing DIscrete Loss (ODIL), robust variational residual minimization, and Isogeometric Analysis (IGA). Instead of neural-network parameterizations of PINNs, the unknown solution is represented by smooth B-spline basis functions, leading to sparse structured Jacobians and efficient Gauss--Newton optimization. We also derive robust residual formulations based on weighted Gram operators, making the loss function related with the true error. The resulting systems inherit locality, sparsity, and approximation-theoretic properties of classical finite element and isogeometric methods while preserving the residual-learning philosophy of scientific machine learning. The proposed methodology is evaluated on several benchmark problems, including Poisson equations, convection-dominated advection--diffusion equations, Helmholtz problems with highly oscillatory solutions, nonlinear Allen--Cahn equations, and inverse Helmholtz parameter identification. Numerical experiments demonstrate orders-of-magnitude speedups compared with PINNs and CRVPINNs while maintaining high accuracy and robustness.

Tomasz Służalec, Marcin Łoś, Askold Vilkha et al.

We explore the possibility of solving Partial Differential Equations (PDEs) using discrete weak formulations. We propose a programming environment for defining a discrete computational domain, introducing discrete functions defined over a set of points, constructing discrete inner products, and introducing discrete weak formulations employing Kronecker delta test functions. Building on this setup, we propose a discrete neural network representation, training the solution function defined over a discrete set of points and employing discrete finite difference derivatives in the automatic differentiation procedures. As a challenging computational model example, we focus on Stokes equations in two-dimensions, defined over a discrete set of points. We train the solution using the discrete weak residual and the Adamax algorithm with discrete automatic differentiation of the discrete gradients. Despite introducing the python environment, we also provide a rigorous mathematical formulation based on discrete weak formulations, proving the well-posedness and robustness of the loss function. The solution of the discrete weak formulations is based on neural network training employing a robust loss function that is related to the true error. In this way, we have a robust control of the numerical error during the training of the neural networks. Besides the Stokes formulation, we also explain the functionality of the proposed library using the Laplace problem formulation.

Leszek Siwik, Maciej Sikora, Natalia Leszczyńska et al.

In this paper, we propose a Physics-Informed Neural Network framework for time-dependent simulations of pollution propagation originating from moving emission sources. We formulate a robust variational framework for the time-dependent advection-diffusion problem and establish the boundedness and inf-sup stability of the corresponding discrete weak formulation. Based on this mathematical foundation, we construct a robust loss function that is directly related to the true approximation error, defined as the difference between the neural network approximation and the (unknown) exact solution. Additionally, a collocation-based strategy is introduced to speed up neural network training. As a case study, we investigate pollution propagation caused by snowmobile traffic in Longyearbyen, Spitsbergen, supported by detailed in-field measurements collected using dedicated sensors. The proposed framework is applied to analyze the effects of thermal inversion on pollutant accumulation. Our results demonstrate that thermal inversion traps dense and humid air masses near the ground, significantly enhancing particulate matter (PM) concentration and worsening local air quality.

Askold Vilkha, Tomasz Służalec, Marcin Łoś et al.

This paper presents the hybrid solver for a $CO_2$ sequestration problem. The solver uses the IGA-ADS (IsoGeometric Analysis Alternating Directions solver) to compute the saturation scalar field update using the explicit method, and CRVPINN (Collocation-based Robust Variational Physics Informed Neural Networks solver) to compute the pressure scalar field. The study focuses on simulating the physical behavior of $CO_2$ in porous structures, excluding chemical reactions. The mathematical model is based on Darcy's Law. The CRVPINN is pretrained on the initial pressure configuration, and the time step pressure updates require only 100 iterations of the Adam method per time step. We compare our hybrid IGA-ADS solver, coupled with the CRVPINN method, with a baseline of the IGA-ADS solver coupled with the MUMPS direct solver. Our hybrid solver is over 3 times faster on a single computational node from the ARES cluster of ACK CYFRONET. Future work includes extensive testing, inverse problem solving, and potential application to $H_2$ storage problems.

Juliusz Wasieleski, Tomasz Służalec, Maciej Woźniak et al.

We develop a wildfire simulation model that evolves the temperature scalar field using an energy balance equation accounting for heat generation, transport, and loss. For these equations, we develop quasi-implicit time integration schemes using direction splitting of the differential operators. We use the Peaceman-Rachford and Strang splitting methods, including the Crank-Nicolson method. Based on these discretizations, we derive variational formulations and explore the Kronecker product structure of the matrices. In the wildfire model, there are some non-linear terms that we treat explicitly. We perform a detailed analysis of how treating these terms affects the stability of the time integration scheme. Namely, we show that a quasi-implicit time integration scheme achieves 10 times higher simulation accuracy. We present two wildfire simulations. The first is a simulation of the 2024 wildfire disaster in the Valparaíso region of Chile. The second one is a simulation of the 2019 wildfire disaster in Las Palmas de Gran Canaria, Spain. We discuss the numerical results and compare them against satellite images and measurement records. We also present a numerical experiment for comparison with the state-of-the-art wildfire simulation model FARSITE. Our sequential code has a linear computational cost of ${\cal O}(N)$. We also present the parallel scalability of the WILDFIRE-IGA-ADS code to illustrate the possibility of running the code on a local workstation.

Marcin Łoś, Tomasz Służalec, Paweł Maczuga et al.

Physics-Informed Neural Networks (PINNs) have been successfully applied to solve Partial Differential Equations (PDEs). Their loss function is founded on a strong residual minimization scheme. Variational Physics-Informed Neural Networks (VPINNs) are their natural extension to weak variational settings. In this context, the recent work of Robust Variational Physics-Informed Neural Networks (RVPINNs) highlights the importance of conveniently translating the norms of the underlying continuum-level spaces to the discrete level. Otherwise, VPINNs might become unrobust, implying that residual minimization might be highly uncorrelated with a desired minimization of the error in the energy norm. However, applying this robustness to VPINNs typically entails dealing with the inverse of a Gram matrix, usually producing slow convergence speeds during training. In this work, we accelerate the implementation of RVPINN, establishing a LU factorization of sparse Gram matrix in a kind of point-collocation scheme with the same spirit as original PINNs. We call out method the Collocation-based Robust Variational Physics Informed Neural Networks (CRVPINN). We test our efficient CRVPINN algorithm on Laplace, advection-diffusion, and Stokes problems in two spatial dimensions.