References
- O. SIMEONE, “A Brief Introduction to Machine Learning for Engineers”; https://arxiv.org/abs/1709.02840 (2017).
- A. LINDHOLM et al., Machine Learning: A First Course for Engineers and Scientists, Cambridge University Press (2022).
- G. E. KARNIADAKIS et al., “Physics-Informed Machine Learning,” Nat. Rev. Phys., 3, 422 (2021).
- J. I. C. VERMAAK et al., “Massively Parallel Transport Sweeps on Meshes with Cyclic Dependencies,” J. Comput. Phys., 425, 109892 (2021); https://doi.org/10.1016/j.jcp.2020.109892.
- M. RAISSI, P. PERDIKARIS, and G. E. KARNIADAKIS, “Physics-Informed Neural Networks: A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations,” J. Comput. Phys., 378, 686 (2019); https://doi.org/10.1016/j.jcp.2018.10.045.
- D. P. KINGMA and J. BA, “Adam: A Method for Stochastic Optimization”; https://arxiv.org/abs/1412.6980 (2015).
- D. C. LIU and J. NOCEDAL, “On the Limited Memory BFGS Method for Large Scale Optimization,” Math. Program., 45, 503 (1989); https://doi.org/10.1007/BF01589116.
- S. WANG, X. YU, and P. PERDIKARIS, “When and Why PINNs Fail to Train: A Neural Tangent Kernel Perspective,” J. Comput. Phys., 449, 110768 (2022); https://doi.org/10.1016/j.jcp.2021.110768.
- S. WANG, H. WANG, and P. PERDIKARIS, “On the Eigenvector Bias of Fourier Feature Networks: From Regression to Solving Multi-Scale PDEs with Physics-Informed Neural Networks,” Comput. Methods Appl. Mech. Eng., 384, 113938 (2021); https://doi.org/10.1016/j.cma.2021.113938.
- M. TANCIK et al., “Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains”; https://arxiv.org/abs/2006.10739 (2020).
- S. WANG, S. SANKARAN, and P. PERDIKARIS, “Respecting Causality Is All You Need for Training Physics-Informed Neural Networks”; https://arxiv.org/abs/2203.07404 (2022).
- A. DAW et al., “Mitigating Propagation Failures in PINNs Using Evolutionary Sampling”; https://arxiv.org/abs/2207.02338 (2022).
- R. MOJGANI, M. BALAJEWICZ, and P. HASSANZADEH, “Lagrangian PINNs: A Causality-Conforming Solution to Failure Modes of Physics-Informed Neural Networks”; https://arxiv.org/abs/2205.02902 (2022).
- E. E. LEWIS and W. F. MILLER, Computational Methods of Neutron Transport, p. 1, Wiley (1984).
- T. R. HILL, “ONETRAN: A Discrete Ordinates Finite Element Code for the Solution of the One-Dimensional Multigroup Transport Equation,” LA-5990-MS, p. 6, Los Alamos National Laboratory (1975).
- T. A. WAREING et al., “Discontinuous Finite Element SN Methods on Three-Dimensional Unstructured Grids,” Nucl. Sci. Eng., 138, 3, 256 (2001); https://doi.org/10.13182/NSE138-256.
- M. W. HACKEMACK and J. C. RAGUSA, “Quadratic Serendipity Discontinuous Finite Element Discretization for Sn Transport on Arbitrary Polygonal Grids,” J. Comput. Phys., 374, 188 (2018); https://doi.org/10.1016/j.jcp.2018.05.032.
- A. G. BAYDIN et al., “Automatic Differentiation in Machine Learning: A Survey,” J. Mach. Learn. Res., 18, 1 (2018).
- M. M. POZULP et al., “Heterogeneity, Hyperparameters, and GPUs: Towards Useful Transport Calculations Using Neural Networks,” LLNL-PROC-819671, Lawrence Livermore National Laboratory (2021).
- W. H. REED, “New Difference Schemes for the Neutron Transport Equation,” Nucl. Sci. Eng., 46, 2, 309 (1971); https://doi.org/10.13182/NSE46-309.
- T. G. GROSSMANN et al., “Can physics-informed neural networks beat the finite element method?” arXiv.2302.04107, 2023.