Abstract
In the field of scientific computing, recent work has studied Physics Informed Neural Networks (PINNs) as a novel computational paradigm for weakly supervised learning-based methods to solve nonlinear partial differential equations (PDEs), bridging progress in classical numerical analysis with recent advances in general machine learning. Traditional numerical methods such as the Finite Difference Method (FDM), the Finite Element Method (FEM) and spectral approaches, which have been around for decades to numerically solve partial differential equations modelling a physical system. However, their performance suffers from limitations such as mesh dependency and high computational cost in higher dimensional domains, and they struggle to represent complicated geometries or sparse data scenarios. Whereas, PINNs exploit the Representational power of Deep Neural Networks (DNNs) to approximate a continuous solution along with physical laws as differential equation constraints that embedded directly in the learning Processions centre around the idea of embedding governing nonlinear PDEs into the loss function of a deep learning model. The network is trained not on labelled data but by minimizing the residuals of the PDE, plus boundary and initial condition constraints. This enables the model to recover physically consistent solutions, even in data-poor or noisy regimes. Moreover, a mesh-free operation feature enables PINNs to address irregular domains and high-dimensions problems fitting the modern computational science missions well.
Nonlinear Partial Differential Equations (PDEs) arise from numerous natural and engineering sciences, such as fluid mechanics, quantum physics and heat transfer; their innate complexities include nonlinearity behaviour together with sensitivity to initial and boundary conditions. This problem is formulated in terms of optimizing generalised versions of deep neural networks (DNN) — such universal function approximates lead to a class of methods called physics-informed neural networks (PINNs). PINNs can take advantage of automatic differentiation to calculate the derivatives required for direct calculation of PDE residuals, making them not in need of a specific discretization scheme. Although PINNs have many potential features, there are a few capabilities they lack. However, problems such as training instability, spectral bias, slow convergence and sensitivity to hyper parameters are under-studied. Alternatively, Even when in the same problem type that is required to minimize multiple loss components; PDE residuals and boundary conditions, arise many issues to serial tough problems. Recent advances, including adaptive loss weighting, domain decomposition strategies, and hybrid modelling using data-driven and physics-based approaches are beginning to address the limitations of current practice.
In this research paper, we provide a review of Physics-Informed Neural Networks for solving nonlinear PDEs in a detailed and high-level way. Fundamentals of nonlinear PDEs, mathematical foundations for PINNs, and optimization dynamics; advances in architecture and comparisons to traditional numeric solvers Moreover, it focuses on real-world applications and discusses current advances and research trends. In this work we aim to bridge the gap in leveraging our understanding into PINNs for scientific computing based on the insights in functional analysis, deep learning theory and computational physics [11].
Keywords
Physics-Informed Neural Networks, Nonlinear Partial Differential Equations, Deep Learning, Scientific Machine Learning, Automatic Differentiation, Variation Methods, Neural Operators, Computational Physics
References
[1] Raise, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686-707. https://doi.org/10.1016/j.jcp.2018.10.045 (Cited by: 11245)
[2] Wang, S., Tang, Y., & Perdikaris, P. (2021). Understanding and mitigating gradient pathologies in physics-informed neural networks. SIAM Journal on Scientific Computing, 43(5), A3055–A3081. https://doi.org/10.1137/20M1318043
[3] Wang, S., Wang, H., & Perdikaris, P. (2021). On the eigenvector bias of Fourier feature networks: From regression to solving partial differential equations. SIAM Journal on Scientific Computing, 43(3), A1766–A1794. https://doi.org/10.1137/20M1339120
[4] Salzmann, V., Martel, J., Bergman, A., Liddell, D., & Wet stein, G. (2020). Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems (Neutrals), 33, 7462-7473.
[5] Wang, S., Yu, X., & Perdikaris, P. (2022). When and why PINNs fail to train: A neural tangent kernel perspective. Journal of Computational Physics, 449, 110768. https://doi.org/10.1016/j.jcp.2021.110768
[6] GU, Y., Cao, X., & Wang, L. (2024). A Residual-Based Adaptive Refinement Physics-Informed Neural Networks (RAR-PINNs) method for fifth-order Kidd equation. Research Gate. https://doi.org/10.13140/RG.2.2.14620.19842
[7] Jag tap, A. D., & Karniadakis, G. E. (2020). Extended Physics-informed Neural Networks (XPINNs): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations. Communications in Computational Physics, 28(5), 2002-2041. https://doi.org/10.4208/cicp.OA-2020-0164
[8] Jag tap, A. D., Kharrazi, E., & Karniadakis, G. E. (2020). Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems. Computer Methods in Applied Mechanics and Engineering, 365, 113028. https://doi.org/10.1016/j.cma.2020.113028
[9] Lu, L., Men, X., Mao, Z., & Karniadakis, G. E. (2021). Deeded: A deep learning library for solving differential equations. SIAM Review, 63(1), 208-228. https://doi.org/10.1137/19M1274067 (Cited by: 1240)
[10] Tannic, M., Srinivasan, P., Mildenhall, B., et al. (2020). Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems (Neutrals), 33, 7537-7547.
[11] Krishnapriyan, A., Thalami, A., Zhen, S., Kirby, R., & Mahoney, M. W. (2021). Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems (Neutrals), 34, 26561-26573.
[12] Cu, S., et al. (2025). Copen: Cognitive Physics-Informed Neural Networks for embracing robustness and generalization. International Conference on Machine Learning (ICML). https://icml.cc/virtual/2025/poster/46458
[13] Shula, N., et al. (2026). Domain Decomposition PINNs: CPINN & XPINN — A Review of Methodologies for Scientific Machine Learning. Emergent Mind.
[14] Martinis, S. (2021). The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers? Frontiers in Big Data, 4. https://doi.org/10.3389/fdata.2021.669097
[15] Jin, X., Cain, S., Li, H., & Karniadakis, G. E. (2021). NSFnets: Navies-Stokes flow nets for solving incompressible Navies-Stokes equations. Journal of Computational Physics, 426, 110014. https://doi.org/10.1016/j.jcp.2020.110014
[16] Zhang, J., et al. (2026). Physics-Informed Neural Networks: A Review of Methodological Evolution, Theoretical Foundations, and Interdisciplinary Frontiers. Applied Sciences (MDPI), 15(14), 8092. https://doi.org/10.3390/app15148092
[17] Cho, M., et al. (2023). Separable PINN (SPINN): Efficient training of physics-informed neural networks. Advances in Neural Information Processing Systems (Neutrals).
[18] Xiang, Z., Pang, W., Zhou, X., & Wen, W. (2022). Self-adaptive loss balanced physics-informed neural networks. Neurocomputing, 496, 11-34.
[19] Wang, S., & Perdikaris, P. (2024). Long-time integration of physics-informed neural networks via curriculum learning. Journal of Computational Physics, 502, 112815.
[20] Redman, A., et al. (2025). Space-time domain decomposition for unsteady multistate problems in XPINNs. Research Gate.
[21] Yang, L., Men, X., & Karniadakis, G. E. (2021). B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with uncertainty quantification. Journal of Computational Physics, 425, 109913.
[22] Liu, D., & Wang, J. (2026). Self-adaptive weighting and sampling for physics-informed neural networks. Journal of Computational Physics (Preprint). https://www.researchgate.net/publication/402962875
[23] MacClenny, L. D., & Braga-Net, U. M. (2020). Self-adaptive physics-informed neural networks. arrive preprint arXiv:2009.04544.
[24] Bishop, A., & Kraus, M. (2021). Multi-objective loss balancing for physics-informed deep learning. arrive preprint arXiv:2110.09813.
[25] Daw, A., et al. (2022). Rethinking the importance of sampling in physics-informed neural networks. Proceedings of the Machine Learning for Engineering Modelling, Simulation, and Design (ML4Eng).
[26] Gaol, Z., et al. (2025). Spectral PINNSformer (S-Former): Mitigating spectral bias via Fourier feature embedding’s and attention-driven decoding. Chat Paper. https://chatpaper.com/paper/197244
[27] Li, Z., et al. (2026). Curriculum-Learned Vanishing Stacked Residual PINNs for Hyperbolic PDE State Reconstruction. Research Gate.
[28] Lu, L., et al. (2021). Learning nonlinear operators via Deponents based on the universal approximation theorem of operators. Nature Machine Intelligence, 3(3), 218-229. (Cited by: 1540)
[29] Wang, Z., et al. (2025). A Gradient-Variance Weighting Physics-Informed Neural Network for Solving Integer and Fractional PDEs. Applied Sciences (MDPI), 15(20), 11137. https://doi.org/10.3390/app152011137
[30] Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., Wang, S., & Yang, L. (2021). Physics-informed machine learning. Nature Reviews Physics, 3(6), 422-440. (Cited by: 2890)