Learning Where the Physics Is: Probabilistic Adaptive Sampling for Stiff PDEs

Physics-informed neural networks (PINNs) offer a mesh-free framework for solving partial differential equations (PDEs), yet training often suffers from gradient pathologies, spectral bias, and poor convergence, especially for problems with strong nonlinearity, sharp gradients, or multiscale features. We propose the Curriculum-Guided Gaussian Mixture Physics-Informed Neural Network (CGMPINN), which integrates Gaussian mixture modeling with dynamic curriculum learning. Specifically, a GMM is periodically fitted to the PDE residual distribution to quantify spatially varying learning difficulty. A smooth curriculum schedule progressively shifts training focus from easy to harder regions, while precision-based variance modulation suppresses unreliable clusters during early optimization. This dual curriculum is governed by a shared curriculum parameter and can be combined with self-adaptive loss balancing. We further establish theoretical guarantees, including sublinear convergence of the gradient norm for the induced time-varying loss, uniform equivalence between the curriculum-weighted and standard PDE losses, and a generalization bound with an explicit weighting-induced bias characterization. Experiments on six benchmark PDEs spanning elliptic, parabolic, hyperbolic, advection-dominated, and nonlinear reaction-diffusion types show that CGMPINN consistently achieves the lowest relative $L_2$ and maximum absolute errors among all compared methods, reducing relative $L_2$ error by up to 97.8% over the standard PINN at comparable cost. Our code is publicly available at https://github.com/Mathematics-Yang/CGMPINN.

Solving partial differential equations (PDEs) with high-frequency solutions remains a central challenge in physics-informed machine learning due to spectral bias -- the tendency of neural networks to learn low-frequency components preferentially. This paper proposes a Frequency Shift Physics-Informed Extreme Learning Machine (FS-PIELM) framework that addresses this limitation through an additive mechanism for weight initialization. Rather than multiplying random weights by a scaling factor, the method translates the mean of the Gaussian weight distribution while keeping the variance fixed at unity, thereby avoiding the variance amplification inherent in scaling-based methods. Two variants are developed: FS-PIELM-L assigns independent frequency magnitudes to individual neurons, while FS-PIELM-G groups neurons for improved robustness. Theoretical analysis shows that the frequency variance under the proposed framework remains bounded and approaches unity regardless of target frequency, in contrast to the quadratic growth of conventional approaches. The method preserves the computational efficiency of extreme learning machines, requiring only a single linear solve. Experiments on seven benchmark problems spanning six equation types -- Helmholtz, wave, Poisson, Klein-Gordon, heat, and advection-diffusion -- on both regular and complex geometries show that the linear variant achieves the best accuracy in six of seven cases, with improvements of one to nearly five orders of magnitude over existing PIELM variants. The code and data accompanying this manuscript will be made publicly available at https://github.com/xgxgnpu/Physics-informed-vibe-coding/tree/main/FS-PIELM.

Partial differential equations (PDEs) play a central role in modeling complex physical, biological, and engineering systems. While traditional numerical solvers are robust, they often incur prohibitive computational costs due to mesh dependencies, whereas recent Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative but frequently suffer from slow convergence and optimization instability. To bridge this gap, this article proposes the Physics-Informed Broad Learning System (PIBLS), a novel backpropagation-free framework that reformulates PDE solving as a direct least-squares optimization. We improved an algorithm within this framework to handle nonlinear PDEs efficiently and provide a rigorous mathematical proof establishing the universal approximation property of PIBLS for these equations. Experiments on linear and nonlinear PDEs demonstrate that PIBLS is one to three orders of magnitude faster than conventional PINNs while achieving significantly higher solution accuracy. This framework provides a computationally efficient paradigm for scientific machine learning, offering a practical, high-speed alternative for real-time simulation and design optimization tasks.