arXiv subjects

Bruno Sixou

Publications and source records attributed to Bruno Sixou.

7 recordsLinked to original sources

arXiv2026

Meta-Learned Basis Adaptation for Parametric Linear PDEs

We propose a hybrid physics-informed framework for solving families of parametric linear partial differential equations (PDEs) by combining a meta-learned predictor with a least-squares corrector. The predictor, termed \textbf{KAPI} (Kernel-Adaptive Physics-Informed meta-learner), is a shallow task-conditioned model that maps query coordinates and PDE parameters to solution values while internally generating an interpretable, task-adaptive Gaussian basis geometry. A lightweight meta-network maps PDE parameters to basis centers, widths, and activity patterns, thereby learning how the approximation space should adapt across the parametric family. This predictor-generated geometry is transferred to a second-stage corrector, which augments it with a background basis and computes the final solution through a one-shot physics-informed Extreme Learning Machine (PIELM)-style least-squares solve. We evaluate the method on four linear PDE families spanning diffusion, transport, mixed advection--diffusion, and variable-speed transport. Across these cases, the predictor captures meaningful physics through localized and transport-aligned basis placement, while the corrector further improves accuracy, often by one or more orders of magnitude. Comparisons with parametric PINNs, physics-informed DeepONet, and uniform-grid PIELM correctors highlight the value of predictor-guided basis adaptation as an interpretable and efficient strategy for parametric PDE solving.

cs.LG↗

arXiv2026

Soft Partition-based KAPI-ELM for Multi-Scale PDEs

Physics-informed machine learning holds great promise for solving differential equations, yet existing methods struggle with highly oscillatory, multiscale, or singularly perturbed PDEs due to spectral bias, costly backpropagation, and manually tuned kernel or Fourier frequencies. This work introduces a soft partition--based Kernel-Adaptive Physics-Informed Extreme Learning Machine (KAPI-ELM), a deterministic low-dimensional parameterization in which smooth partition lengths jointly control collocation centers and Gaussian kernel widths, enabling continuous coarse-to-fine resolution without Fourier features, random sampling, or hard domain interfaces. A signed-distance-based weighting further stabilizes least-squares learning on irregular geometries. Across eight benchmarks--including oscillatory ODEs, high-frequency Poisson equations, irregular-shaped domains, and stiff singularly perturbed convection-diffusion problems-the proposed method matches or exceeds the accuracy of state-of-the-art Physics-Informed Neural Network (PINN) and Theory of Functional Connections (TFC) variants while using only a single linear solve. Although demonstrated on steady linear PDEs, the results show that soft-partition kernel adaptation provides a fast, architecture-free approach for multiscale PDEs with broad potential for future physics-informed modeling. For reproducibility, the reference codes are available at https://github.com/vikas-dwivedi-2022/soft_kapi

cs.LG↗

arXiv2025

Gated X-TFC: Soft Domain Decomposition for Forward and Inverse Problems in Sharp-Gradient PDEs

Physics-informed neural networks (PINNs) and related methods struggle to resolve sharp gradients in singularly perturbed boundary value problems without resorting to some form of domain decomposition, which often introduce complex interface penalties. While the Extreme Theory of Functional Connections (X-TFC) avoids multi-objective optimization by employing exact boundary condition enforcement, it remains computationally inefficient for boundary layers and incompatible with decomposition. We propose Gated X-TFC, a novel framework for both forward and inverse problems, that overcomes these limitations through a soft, learned domain decomposition. Our method replaces hard interfaces with a differentiable logistic gate that dynamically adapts radial basis function (RBF) kernel widths across the domain, eliminating the need for interface penalties. This approach yields not only superior accuracy but also dramatic improvements in computational efficiency: on a benchmark one dimensional (1D) convection-diffusion, Gated X-TFC achieves an order-of-magnitude lower error than standard X-TFC while using 80 percent fewer collocation points and reducing training time by 66 percent. In addition, we introduce an operator-conditioned meta-learning layer that learns a probabilistic mapping from PDE parameters to optimal gate configurations, enabling fast, uncertainty-aware warm-starting for new problem instances. We further demonstrate scalability to multiple subdomains and higher dimensions by solving a twin boundary-layer equation and a 2D Poisson problem with a sharp Gaussian source. Overall, Gated X-TFC delivers a simple alternative alternative to PINNs that is both accurate and computationally efficient for challenging boundar-layer regimes. Future work will focus on nonlinear problems.

cs.LG↗

arXiv2025

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

Physics-informed machine learning frameworks such as Physics-Informed Neural Networks (PINNs) and Physics-Informed Extreme Learning Machines (PI-ELMs) have shown great promise for solving partial differential equations (PDEs) but struggle with localized sharp gradients and singularly perturbed regimes, PINNs due to spectral bias and PI-ELMs due to their single-shot, non-adaptive formulation. We propose the Kernel-Adaptive Physics-Informed Extreme Learning Machine (KAPI-ELM), which performs Bayesian optimization over a low-dimensional, physically interpretable hyperparameter space governing the distribution of Radial Basis Function (RBF) centers and widths. This converts high-dimensional weight optimization into a low-dimensional distributional search, enabling targeted kernel refinement in regions with sharp gradients while also improving baseline solutions in smooth-flow regimes by tuning RBF supports. KAPI-ELM is validated on benchmark forward and inverse problems (1D convection-diffusion and 2D Poisson) involving PDEs with sharp gradients. It accurately resolves steep layers, improves smooth-solution fidelity, and recovers physical parameters robustly, matching or surpassing advanced methods such as the extended Theory of Functional Connections (X-TFC) with nearly an order of magnitude fewer tunable parameters. An extension to nonlinear problems is demonstrated by a curriculum-based solution of the steady Navier-Stokes equations via successive linearizations, yielding stable solutions for benchmark lid-driven cavity flow up to Re=100. These results indicate that KAPI-ELM provides an efficient and unified approach for forward and inverse PDEs, particularly in challenging sharp-gradient regimes.

cs.LG↗

arXiv2025

Curriculum Learning-Driven PIELMs for Fluid Flow Simulations

This paper presents two novel, physics-informed extreme learning machine (PIELM)-based algorithms for solving steady and unsteady nonlinear partial differential equations (PDEs) related to fluid flow. Although single-hidden-layer PIELMs outperform deep physics-informed neural networks (PINNs) in speed and accuracy for linear and quasilinear PDEs, their extension to nonlinear problems remains challenging. To address this, we introduce a curriculum learning strategy that reformulates nonlinear PDEs as a sequence of increasingly complex quasilinear PDEs. Additionally, our approach enables a physically interpretable initialization of network parameters by leveraging Radial Basis Functions (RBFs). The performance of the proposed algorithms is validated on two benchmark incompressible flow problems: the viscous Burgers equation and lid-driven cavity flow. To the best of our knowledge, this is the first work to extend PIELM to solving Burgers' shock solution as well as lid-driven cavity flow up to a Reynolds number of 100. As a practical application, we employ PIELM to predict blood flow in a stenotic vessel. The results confirm that PIELM efficiently handles nonlinear PDEs, positioning it as a promising alternative to PINNs for both linear and nonlinear PDEs.

cs.LG↗

arXiv2022

Towards the optimization of ballistics in proton therapy using genetic algorithms: implementation issues

The dose delivered to the planning target volume by proton beams is highly conformal, sparing organs at risk and normal tissues. New treatment planning systems adapted to spot scanning techniques have been recently proposed to simultaneously optimize several fields and thus improve dose delivery. In this paper, we investigate a new optimization framework based on a genetic algorithm approach. This tool is intended to make it possible to explore new schemes of treatment delivery, possibly with future enhanced technologies. The optimization framework is designed to be versatile and to account for many degrees of freedom, without any {\it a priori} technological constraint. To test the behavior of our algorithm, we propose in this paper, as an example, to optimize beam fluences, target points and irradiation directions at the same time. The proposed optimization routine takes typically into account several thousands of spots of fixed size. The evolution is carried out by the three standard genetic operators: mutation, crossover and selection. The figure-of-merit (or fitness) is based on an objective function relative to the dose prescription to the tumor and to the limits set for organs at risk and normal tissues. Fluence optimization is carried out via a specific scheme based on a plain gradient with analytical solution. Several specific genetic algorithm issues are addressed: (i) the mutation rate is tuned to balance the search and selection forces, (ii) the initial population is selected using a bootstrap technique and (iii) to scale down the computation time, dose calculations are carried out with a fast analytical ray tracing method and are multi-threaded. In this paper implementation issues of the optimization framework are thoroughly described. The behavior of the proposed genetic algorithm is illustrated in both elementary and clinically-realistic test cases.

physics.med-ph↗

arXiv2022

Variable length genetic algorithm with continuous parameters optimization of beam layout in proton therapy

Proton therapy is a modality in fast development. Characterized by a maximum dose deposition at the end of the proton trajectory followed by a sharp fall-off, proton beams can deliver a highly conformal dose to the tumor while sparing organs at risk and surrounding healthy tissues. New treatment planning systems based on spot scanning techniques can now propose multi-field optimization. However, in most cases, this optimization only processes the field fluences whereas the choice of ballistics (field geometry) is left to the oncologist and medical physicist. In this work, we investigate a new optimization framework based on a genetic approach. This tool is intended to explore new irradiation schemes and to evaluate the potential of actual or future irradiation systems. We propose to optimize simultaneously the target points and beam incidence angles in a continuous manner and with a variable number of beams. No \textit{a priori} technological constraints are taken into account, \textit{i.e.}~the beam energy values, incidence directions and target points are free parameters. The proposed algorithm is based on a modified version of classical genetic operators: mutation, crossover and selection. We use the real coding associated with random perturbations of the parameters to obtain a continuous variation of the potential solutions. We also introduce a perturbation in the exchange points of the crossover to allow variations of the number of beams. These variations are controlled by introducing a beam fluence lower limit. In this paper, we present a complete description of the algorithm and of its behaviour in an elementary test case. The proposed method is finally assessed in a clinically-realistic test case.

physics.med-ph↗