Experiments show that FC-VPINN achieves approximately one-order-of-magnitude lower prediction errors than the traditional PINN and reduces memory usage to 40% of that required by the baseline, demonstrating improved accuracy and computational efficiency in multidimensional problems.
Abstract
Traditional Physics-Informed Neural Networks (PINNs) incorporate partial differential equations (PDEs), boundary conditions, and initial conditions into model training. However, their application to multidimensional problems may involve high computational costs associated with high-order automatic differentiation and increasingly demanding sampling requirements as dimensionality grows, potentially limiting the accuracy of solution approximation. To address these issues, we propose a Functional Constraint-based Variational Physics-Informed Neural Network (FC-VPINN) for solving PDEs admitting self-adjoint or weighted self-adjoint representations. It reformulates strong-form PDE constraints as corresponding variational constraints, improving mathematical interpretability while reducing reliance on high-order automatic differentiation. The stationarity condition of the constructed quadratic variational functional is mathematically equivalent to the governing equation under the prescribed conditions. For the second-order PDEs considered in this work, the resulting variational objective involves at most first-order derivatives of the network output, thereby reducing the derivative order required during training. Boundary and initial conditions are imposed through corresponding loss terms, without requiring a predefined set of test functions. Experiments on two-dimensional advection-diffusion, three-dimensional diffusion, and three-dimensional Poisson equations show that FC-VPINN achieves approximately one-order-of-magnitude lower prediction errors than the traditional PINN and reduces memory usage to 40% of that required by the baseline, demonstrating improved accuracy and computational efficiency in multidimensional problems.
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.
Pinki Khatun, M. Sajid, Abhinav Jha et al.· 0 citations
In this study, we propose a Physics-Informed Neural Networks (PINNs) framework that incorporates Tikhonov regularization to solve terminal-state tracking optimal control constrained by parabolic partial differential equations (PDEs). This problem is inherently ill-posed, as infinitely many distributed controls may drive the system to the same desired state, so the regularization guides the optimizer toward the minimum-energy control, restoring numerical stability and yielding a smooth, physically meaningful solution. On the theoretical side, we establish a consistency result showing that PINNs minimizers nearly attain the continuous regularized objective under residual and quadrature approximation assumptions, and a novel error estimate that bounds the deviation of the learned control from the minimum-energy solution in terms of the PINNs training residuals and the regularization parameter. Numerical experiments on the linear heat equation and the nonlinear Burgers'equation demonstrate that the regularized PINNs framework accurately achieves the target terminal state while producing controls with significantly lower energy and smoother profiles compared to unregularized baselines.
Partial differential equations (PDEs) are key tools for modeling continuous physical processes, but traditional solvers are costly for high‐dimensional problems with complex boundaries. Existing neural network solvers usually add physical constraints only as loss terms, which limits physical constraint embedding learning and can cause local violations. To address this issue, this paper proposes a Physics Constraint‐Guided Network (PCGN) for deep PDE solving. Its main novelty is to introduce physical information at three levels: feature representation, optimization, and output correction. First, governing equations, boundary conditions, and initial conditions are encoded into propagatable constraint features, and neighborhood propagation improves local consistency. Second, adaptive residual balancing adjusts different constraint terms, reducing instability from uneven residual scales. Third, a differentiable constraint projection layer corrects predictions toward feasible solutions. Experiments on Burgers' equation and Darcy flow show that PCGN achieves lower absolute and relative errors than existing deep learning solvers, while improving training stability and physical consistency.
This work introduces a variational boosting framework in which solutions are constructed additively in function space and separates global nonlinear refinement into a sequence of well-conditioned subproblems while preserving the full variational structure of the operator.
The proposed SCV-PINN provides a robust and generalized extension of standard PINNs for complex-valued, multiscale, oscillatory, high-dimensional, and real-valued nonlinear PDEs.
Biswanath Barman, Rajendra K. Ray, Debdeep Chatterjee· 0 citations
The Physics-Informed Stochastic Configuration Machine is proposed, a novel backpropagation-free framework for both forward and inverse problems in differential equations that achieves high-fidelity predictive accuracy and robust parameter identification while accelerating the training process by orders of magnitude compared to standard PINNs.
Yueze Song, Zhongzhe Chen, Lihui Cen et al.· 0 citations