LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries

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.

Physics-informed neural networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by embedding governing physical laws into deep neural networks. However, their reliance on computationally expensive gradient-based optimization and deep architectures often results in slow training, high computational cost, and limited scalability. In this work, we propose a novel physics-informed broad learning system (PI-BLS), the first physics-informed learning framework based on broad RdNNs. The proposed formulation embeds the governing differential operator and the associated initial and boundary constraints directly into a linear output-layer optimization problem, thereby replacing nonlinear gradient-based training with a deterministic least-squares solution obtained via the pseudoinverse. Consequently, the entire learning process is reduced to a single linear optimization stage while preserving the underlying physical constraints. As a result, PI-BLS offers an efficient learning paradigm for a physics-informed learning framework for solving PDEs that eliminates iterative backpropagation while preserving the underlying physical constraints. Experimental results on representative forward PDE benchmarks demonstrate that PI-BLS achieves competitive and often superior performance with reduced training time and model parameters compared with conventional PINNs.

Physics-informed neural networks (PINNs) provide an effective way to solve partial differential equations (PDEs) by embedding physical principles into the learning process. However, the conventional PINN formulation, in which all constraints are imposed as soft penalty terms within a composite loss, often exhibits slow convergence, sensitivity to loss weight scaling, and inaccurate boundary enforcement due to poor conditioning of the optimization landscape. To address these limitations, this study proposes a unified hard--soft physics--informed neural network (HSPINN) with adaptive loss weighting. In this framework, Dirichlet and periodic boundary conditions are enforced exactly by construction through analytical or polynomial lifting, masking functions, and periodic feature mappings, while the governing PDE residuals, Neumann fluxes, and initial conditions are treated as soft constraints. An inverse-share softmax strategy dynamically balances the relative importance of individual loss components during training, eliminating manual penalty tuning and improving gradient stability. This formulation ensures boundary admissibility throughout optimization and enhances convergence efficiency and numerical robustness. Applications to representative elliptic (Poisson), parabolic (Burgers), and hyperbolic (convection with periodic boundaries) problems demonstrate that HSPINN consistently achieves faster convergence, higher accuracy, and greater stability than conventional PINNs, establishing a general and scalable foundation for physics-constrained deep learning across science and technology.