Niroomandi, S., Alfaro, I., Gonzalez, D., Cueto, E., Chinasta, F. Real-time simulation of surgery using low-order modeling and X-FEM technology. internal. J. Number. Method Biomed. engineering 28574–588 (2012).
Google Scholar
Kuiper, W., Milde, A., Volkwein, S. Reduced Order Modeling (ROM) for Simulation and Optimization: Powerful Algorithms as a Key Enabler for Scientific Computing (Springer, 2018).
Barnett, D.S. Finite element analysis: from concept to application (Addison-Wesley Pub. Co., 1987).
Reddy, J. Overview of finite element method (McGraw-Hill, 1993).
Raissi, M., Perdicaris, P. & Karniadakis, GE Physics-based deep learning (part i): data-driven solutions of nonlinear partial differential equations. arXiv preprint arXiv:1711.10561 (2017).
Raissi, M., Perdicaris, P. & Karniadakis, GE Physics-based neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Compute. Physics. 378686–707 (2019).
Google Scholar
Jenkiewicz, OC & Taylor, RL. Finite element method: solid mechanicsvol. 2 (Butterworth-Heinemann, 2000).
Meethal, RE et al. Neural networks enhanced with finite element methods for forward and inverse problems. Advanced model. Simul. engineering science. 106 (2023).
Google Scholar
Dissanayake, MG & Phan-Thien, N. Neural network-based approximations for solving partial differential equations. common. numbers. method engineering 10195–201 (1994).
Google Scholar
Kharazmi, E., Zhang, Z. & Karniadakis, GE Variational physics-based neural networks for solving partial differential equations. arXiv preprint arXiv:1912.00873 (2019).
Daubechies, I., DeVore, R., Foucart, S., Hanin, B. & Petrova, G. Nonlinear approximations and (deep) Relu networks. Structure approx. 55127–172 (2022).
Google Scholar
Mhaskar, HN & Poggio, T. Function approximation with deep networks. arXiv preprint arXiv:1905.12882 (2019).
Jagtap, A.D., Kharazmi, E. & Karniadakis, G.E. Conservative physics-based neural networks over discrete domains of conservation laws: Applications to forward and inverse problems. Calculate. Applying the method. Mecha. engineering 365113028 (2020).
Google Scholar
Kharazmi, E., Zhang, Z. & Karniadakis, GE hp-VPINN: Variational physics information neural network with domain decomposition. Calculate. Applying the method. Mecha. engineering 374113547 (2021).
Google Scholar
Sunil, P. & Sills, RB FE-PINNS: Finite element-based physically informed neural networks for surrogate modeling. arXiv preprint arXiv:2412.07126 (2024).
Raissi, M. & Karniadakis, G. E. Hidden physical models: Machine learning for nonlinear partial differential equations. J. Compute. Physics. 357125–141. https://doi.org/10.1016/j.jcp.2017.11.039 (2017).
Google Scholar
Raissi, M., Perdicaris, P. & Karniadakis, GE Physics-based neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Compute. Physics. 378686–707. https://doi.org/10.1016/j.jcp.2018.10.045 (2019).
Google Scholar
McClenny, LD & Braga-Neto, UM Self-adaptive physics-informed neural networks. J. Compute. Physics. 474111722 (2023).
Google Scholar
Abueidda, DW, Lu, Q. & Koric, S. A deep learning method for 3D solid mechanics based on meshless physics. internal. J. Number. Female. engineering 1227182–7201 (2021).
Google Scholar
Wang, L., Liu, G., Wang, G. & Zhang, K. M-PINN: A mesh-based physics-informed neural network for linear elasticity problems in solid mechanics. internal. J. Number. Female. engineering 125e7444 (2024).
Google Scholar
Wang, S., Teng, Y. & Perdicaris, P. Understanding and mitigating gradient pathology in physics-based neural networks. arXiv preprint arXiv:2001.04536 (2020).
Fuks, O. & Tchelepi, H.A. Physics limitations influenced machine learning of nonlinear two-phase transport in porous media. J. Mach. learn. model. Calculate.1 (2020).
Liu, C. & Wu, H. cv-PINN: Efficient learning of variational physics-based neural networks with domain decomposition. Extreme mecha. Let. 63102051 (2023).
Google Scholar
Anandh, T., Ghose, D., Jain, H., Ganesan, S. FastVPINN: Tensor-driven acceleration of VPINN for complex geometries. arXiv preprint arXiv:2404.12063 (2024).
Anand, T. et al. An efficient hp variational PINN framework for incompressible Navier-Stokes equations. arXiv preprint arXiv:2409.04143 (2024).
Abda, M., Hamedi, M., Piollet, E., Blake, C. & Gosselin, FP Finite element neural network methods: A one-dimensional study. next resolution 2100885. https://doi.org/10.1016/j.nexres.2025.100885 (2025).
Google Scholar
Yu, B. et al. Deep Ritz Method: A deep learning-based numerical algorithm for solving variational problems. common. Mathematics. statistics 61–12 (2018).
Google Scholar
Samaniego, E. et al. An energetic approach to solving partial differential equations in computational mechanics with machine learning: Concepts, implementation, and applications. Calculate. Applying the method. Mecha. engineering 362112790 (2020).
Google Scholar
Nguyen-Thanh, VM, Zhuang, X. & Rabczuk, T. Deep energy method for finite deformation hyperelasticity. EUR. J. Mech.-A/Solid 80103874 (2020).
Google Scholar
Fuhg, JN & Bouklas, N. A mixed deep energy method for resolving lumped features in finite strain hyperelasticity. J. Compute. Physics. 451110839 (2022).
Google Scholar
Dong, Y., Liu, T., Li, Z. & Qiao, P. DeepFEM: a new element-based deep learning approach for solving nonlinear partial differential equations in computational solid mechanics. J.Eng.Mecha. 14904022102 (2023).
Google Scholar
Wang, X., Yin, Z.-Y., Wu, W., Zhu, H.-H. Neural network augmented differentiable finite element method for boundary value problems. internal. J. Mech. Science. 285109783 (2025).
Google Scholar
Cai, S., Wang, Z., Wang, S., Perdicaris, P. & Karniadakis, GE Physics-based neural networks for heat transfer problems. J. Heat transfer 143060801 (2021).
Google Scholar
Zhang, N., Xu, K., Yin, Z.Y., and Li, K.-Q. & Jin, Y.-F. A finite element integrated neural network framework for elastic and elastoplastic solids. Calculate. Applying the method. Mecha. engineering 433117474 (2025).
Google Scholar
Abadi, M. et al. Tensorflow: Large-scale machine learning on heterogeneous distributed systems. arXiv preprint arXiv:1603.04467 (2016).
Xu, R., Zhang, D., Rong, M. & Wang, N. Weak-form theory-guided neural network (TgNN-wf) for deep learning of subsurface single- and two-phase flows. J. Compute. Physics. 436110318 (2021).
Google Scholar
Rantangen, H.P. & Mardal, K.-A. Introduction to numerical methods for variational problemsvol. 21 (Springer Nature, 2019).
Bochev, MD & Gunzberger, MD least squares finite element methodvol. 166 (Springer Science & Business Media, 2009).
Burgers, JM A mathematical model to explain the theory of turbulence. Advanced applied mechanics. 1171–199 (1948).
Google Scholar
Kingma, DP Adam: Stochastic optimization techniques. arXiv preprint arXiv:1412.6980 (2014).
Liu, D. & Wang, Y. A dual-dimer method for training physically constrained neural networks on minimax architectures. Neural network. 136112–125 (2021).
Google Scholar
Liu, DC and Nocedal, J. On limited memory BFGS methods for large-scale optimization. Mathematics. program. 45503–528 (1989).
Google Scholar
Basdevant, C. et al. Spectral solution and finite difference decomposition of the Berger equation. Calculate. fluid 1423–41 (1986).
Google Scholar
Orlandi, P. Fluid flow phenomena: Numerical toolkitvol. 55 (Springer Science & Business Media, 2012).
Jagtap, A.D., Kawaguchi, K. & Karniadakis, G.E. Adaptive activation functions accelerate convergence in physics-based deep neural networks. J. Compute. Physics. 404109136 (2020).
Google Scholar
Xiao, Y. et al. A least-squares finite difference-based physics-based neural network for steady incompressible flows. Calculate. Mathematics. application 17533–48 (2024).
Google Scholar
Marchi, CH, Suero, R. & Araki, LK Lid-driven square cavity flow: Numerical solution using a 1024 x 1024 grid. J. Blaz Social Mecha. Science. engineering 31186–198 (2009).
Google Scholar
Ghia, U., Ghia, KN & Shin, C. High-resolution solutions for incompressible flows using Navier-Stokes equations and multigrid methods. J. Compute. Physics. 48387–411 (1982).
Google Scholar
Botella, O. and Peyret, R. Benchmark spectral results for lid-driven cavity flow. Calculate. fluid 27421–433 (1998).
Google Scholar
Alphonius, A. et al. Lethe 1.0: Open source, high-performance, high-order computational fluid dynamics software for single-phase and multiphase flows. Available at SSRN 5090483 (2025).
Chandan, K. et al. Physics-based Hermitian neural network for wet porous fins under local thermal nonequilibrium conditions: Application of clique polynomial method. EUR. Physics. J.Specification top. 1-21 (2024).
Chandan, K. et al. Radiative heat transfer analysis of concave porous fins under local thermal nonequilibrium conditions: Application of clique polynomial method and physics-based neural network. applied mathematics. Mecha. 451613–1632 (2024).
Google Scholar
Kumar, C. et al. Physics-based machine learning predictions for thermal analysis in convective-radiative concave fins with periodic boundary conditions. Zam-J. Applied Mathematics. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 104e202300712 (2024).
Google Scholar
Chandan, K. et al. Physics-based optimized neural network for analyzing the radiative-convective thermal performance of tilted wavy porous fins. case stud. thermal engineering 64105423 (2024).
Google Scholar
Oommen, V. & Srinivasan, B. Solving inverse heat transfer problems without surrogate models: A fast, data-sparse, physics-based neural network approach. J. Compute. Information science. engineering twenty two041012 (2022).
Google Scholar
MayaHTT. TMG heat flow solver. https://help.mayahtt.com/tmg/index.html (2026). Accessed in 2026.
Liu, L. et al. Discontinuity computing using physics-based neural networks. J.Sci.Calculate. 9822 (2024).
Google Scholar
Jusène, C. & Remacre, J.-F. Gmsh: 3D finite element mesh generator with built-in pre- and post-processing capabilities. internal. J. Number. Female. engineering 791309–1331 (2009).
Google Scholar
