Intelligent solution method integrating diverse physics loss functions for solving partial differential equations

PDEs (partial differential equations) serve as fundamental mathematical models in diverse scientific and engineering disciplines, including fluid dynamics, heat transfer, electromagnetics, and quantum mechanics. Despite their broad applicability, traditional numerical methods for solving PDEs—such as finite difference, finite element, and spectral methods—often suffer from high computational costs, meshing complexity, and scalability limitations, especially when applied to high-dimensional or real-time simulation scenarios. These challenges hinder the efficient deployment of PDE-based models in practical applications. In recent years, PINNs have emerged as a promising mesh-free, data-efficient framework for solving PDEs by embedding physical laws directly into the training of deep neural networks. However, the accuracy and convergence speed of standard PINNs remain limited due to the difficulty in balancing multiple loss terms, insufficient representation capacity, and weak enforcement of physical constraints. To address these limitations, this study aims to enhance the accuracy and robustness of PINN-based solvers through structural innovation and improved physical regularization. The primary objective is to develop novel neural network architectures that more effectively integrate multi-source physical information, thereby achieving significantly higher solution accuracy compared to conventional PINN approaches.

This work proposes two advanced PINN-based frameworks: EmPINN and DL-PINN. In EmPINN, a novel network architecture is introduced that incorporates a dimension-expanding mechanism combined with residual connections. Specifically, the input features are mapped into a higher-dimensional latent space through a learnable transformation, enabling the network to capture more complex nonlinear mappings inherent in PDE solutions. The residual connections further facilitate gradient flow and improve training stability. In DL-PINN, this dimension expansion is integrated with a multi-physics loss strategy that simultaneously enforces multiple forms of physical consistency. Beyond the standard PDE residual loss, DL-PINN introduces gradient-enhanced terms that penalize deviations in first- and higher-order derivatives, thus improving the smoothness and physical plausibility of the solution. Additionally, a variational physics-informed loss is incorporated, derived from the weak formulation of the PDE, which provides complementary regularization and enhances robustness, particularly in regions with sharp gradients or discontinuities. Both models are trained using a unified loss function that balances the data fidelity, PDE residuals, boundary/initial conditions, and the additional physics-based terms through adaptive weighting strategies. The training process relies solely on collocation points sampled from the domain and boundaries, without requiring labeled solution data, maintaining the unsupervised nature of PINNs.

The proposed methods are evaluated on two canonical PDE benchmarks: the Poisson equation and the Burgers’ equation, both of which exhibit distinct challenges such as nonlinearity, shock formation, and multi-scale behavior. Quantitative comparisons are conducted against standard PINN and other enhanced variants under identical network sizes and training budgets. Experimental results demonstrate that both EmPINN and DL-PINN achieve superior accuracy, with relative L2 errors reduced by one to two orders of magnitude across all test cases. Notably, DL-PINN exhibits the best performance due to its comprehensive integration of gradient enhancement and variational principles, showing particular strength in capturing fine-scale solution features and maintaining stability over long-time simulations. Ablation studies confirm the individual contributions of dimension expansion, residual connections, and multi-physics loss components, validating the effectiveness of each design choice. Furthermore, the models show improved convergence rates, requiring fewer training iterations to reach high-accuracy solutions, which indicates enhanced optimization dynamics.

This study presents two innovative deep learning frameworks—EmPINN and DL-PINN—for solving partial differential equations with significantly improved accuracy and robustness. By introducing dimension expansion and multi-physics loss integration, the proposed methods more effectively leverage the expressive power of neural networks while enforcing physical consistency from multiple perspectives. The results confirm that the fusion of structural enhancements and diversified physical constraints leads to substantial performance gains over traditional PINNs. These advancements contribute to the development of efficient, accurate, and scalable surrogate solvers for complex PDE systems. While the current work focuses on proof-of-concept demonstrations, future research will investigate the interpretability of such models and explore automated neural architecture design tailored to specific PDE types. With ongoing advances in deep learning, PINN-based methods hold great promise for enabling fast, precise, and adaptive solutions to a wide range of scientific computing problems.