Learning universal approximations for partial differential equations with Physics-Informed Broad Learning System

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.

Temporal Smoothness Doubly Robust Learning for Debiased Knowledge Tracing

Knowledge Tracing (KT) is fundamental to intelligent education systems, yet relies on educational logs that are selectively observed. The non-random nature of exercise recommendations and student choices inevitably induces severe selection bias. Most existing KT methods neglect this issue, training on observed logs using standard empirical risk, which yields biased mastery estimates and accumulates errors in subsequent recommendations. To address this, we introduce a doubly robust (DR) formulation for KT that integrates a propensity model with an error imputation model, theoretically guaranteeing unbiasedness if either model is accurate. Beyond unbiasedness, in the sequential setting of KT, we identify that the estimator's performance is compromised by variance-dependent stochastic deviations that accumulate over time, thereby causing training instability and limiting performance. To mitigate this, we derive a generalization bound that explicitly characterizes the impact of estimator variance and identifies temporal smoothness as a key factor in controlling it. Building on these theoretical insights, we propose the Temporal Smoothness Doubly Robust (TSDR) framework. TSDR jointly optimizes the KT predictor and the imputation model with a smoothness regularizer, effectively reducing variance while preserving the unbiasedness guarantee of DR. Experiments on multiple real-world benchmarks demonstrate that TSDR consistently enhances various state-of-the-art KT backbones, underscoring the vital role of principled bias correction in KT.