Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
119 tokens/sec
GPT-4o
56 tokens/sec
Gemini 2.5 Pro Pro
43 tokens/sec
o3 Pro
6 tokens/sec
GPT-4.1 Pro
47 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Physics Informed Neural Networks for Simulating Radiative Transfer (2009.13291v3)

Published 25 Sep 2020 in cs.LG and stat.ML

Abstract: We propose a novel machine learning algorithm for simulating radiative transfer. Our algorithm is based on physics informed neural networks (PINNs), which are trained by minimizing the residual of the underlying radiative tranfer equations. We present extensive experiments and theoretical error estimates to demonstrate that PINNs provide a very easy to implement, fast, robust and accurate method for simulating radiative transfer. We also present a PINN based algorithm for simulating inverse problems for radiative transfer efficiently.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (48)
  1. A. R. Barron. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inform. Theory., 39(3):930–945, 1993.
  2. R. E. Caflisch. Monte carlo and quasi-monte carlo methods. Acta Numerica, 7:1–49, 1998.
  3. J. I. Castor. Radiation hydrodynamics. Cambridge University Press, 2004.
  4. Radiation transfer in an anisotropically scattering plane-parallel medium with space-dependent albedo ω𝜔\omegaitalic_ω(x). Journal of Quantitative Spectroscopy and Radiative Transfer, 34(3):263–270, 1985.
  5. Physics-informed neural networks for inverse problems in nano-optics and metamaterials. Preprint, available from arXiv:1912.01085, 2019.
  6. Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Communications in Mathematics and Statistics, 5(4):349–380, 2017.
  7. De novo structure prediction with deep-learning based scoring. Annual Review of Biochemistry, 77(363-382):6, 2018.
  8. R. Fletcher. Practical methods of optimization. John Wiley and sons, 1987.
  9. M. Frank. Approximate models for radiative transfer. Bull. Inst. Math. Acad. Sinica (New Series), 2:409–432, 2007.
  10. Deep learning. MIT press, 2016.
  11. F. Graziani. The prompt spectrum of a radiating sphere: Benchmark solutions for diffusion and transport. In Computational methods in transport: verification and validation, LNCSE-62, pages 151–167. Springer, 2008.
  12. K. Grella. Sparse tensor approximation for radiative transport. PhD thesis, ETH Zurich, 2013.
  13. K. Grella and C. Schwab. Sparse tensor spherical harmonics approximation in radiative transfer. J. Comput. Phys., 230:8452–8473, 2011.
  14. Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences, 115(34):8505–8510, 2018.
  15. 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, 2020.
  16. Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. Journal of Computational Physics, 404:109126, 2020.
  17. 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, 2020.
  18. G. Kanschat and et. al. Numerical methods in multi-dimensional radiative transfer. Springer, 2008.
  19. Enabling high fidelity neutron transport simulations on petascle architectures. In Proceedings of the Conference on High Performance Computing Networking, Storage, and Analysis, volume 67, Portland, Oregon, 2009.
  20. D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. In 3rd International Conference on Learning Representations, ICLR 2015, 2015.
  21. Neural-network methods for bound- ary value problems with irregular boundaries. IEEE Transactions on Neural Networks, 11:1041–1049, 2000.
  22. Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks, 9(5):987–1000, 1998.
  23. K. D. Lathrop. Ray effects in discrete ordinates equations. Nucl. Sci. Eng, 32:357, 1968.
  24. Deep learning. Nature, 521(7553):436–444, 2015.
  25. B-pinns: Bayesian physics-informed neural networks for forward and inverse pde problems with noisy data. Preprint, available from arXiv:2003.06097, 2020.
  26. Deepxde: A deep learning library for solving differential equations. Preprint, available from arXiv:1907.04502, 2019.
  27. Iterative surrogate model optimization (ismo): An active learning algorithm for pde constrained optimization with deep neural networks. Preprint, available as arXiv:2008.05730, 2020.
  28. Deep learning observables in computational fluid dynamics. Journal of Computational Physics, page 109339, 2020.
  29. Physics-informed neural networks for high-speed flows. Computer Methods in Applied Mechanics and Engineering, 360:112789, 2020.
  30. S. Mishra and R. Molinaro. Estimates on the generalization error of physics informed neural networks (pinns) for approximating a class of inverse problems for pdes. Preprint, available from arXiv:2007.01138, 2020.
  31. S. Mishra and R. Molinaro. Estimates on the generalization error of physics informed neural networks (pinns) for approximating pdes. Preprint, available from arXiv:2006:16144v1, 2020.
  32. S. Mishra and T. K. Rusch. Enhancing accuracy of deep learning algorithms by training with low-discrepancy sequences. Preprint, available as arXiv:2005.12564, 2020.
  33. M. F. Modest. Radiative heat transfer. Elsevier, 2003.
  34. M. F. Modest and J. Yang. Elliptic pde formulation and boundary conditions of the spherical harmonics method of arbitrary order for general three-dimensional geometries. Journal of Quantitative Spectroscopy and Radiative Transfer, 109:1641–1666, 2008.
  35. Foundations of machine learning. MIT press, 2018.
  36. fpinns: Fractional physics-informed neural networks. SIAM journal of Scientific computing, 41:A2603–A2626, 2019.
  37. Automatic differentiation in pytorch. In Workshop Proceedings of Neural Information Processing Systems, 2017.
  38. J. Pontaza and J. Reddy. Least-squares finite element formulations for one-dimensional radiative transfer. Journal of Quantitative Spectroscopy and Radiative Transfer, 95(3):387–406, 2005.
  39. M. Raissi and G. E. Karniadakis. Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357:125–141, 2018.
  40. 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, 2019.
  41. Hidden fluid mechanics: A navier-stokes informed deep learning framework for assimilating flow visualization data. arXiv preprint arXiv:1808.04327, 2018.
  42. Radiative transfer with finite elements-i. basic method and tests. Astronomy & Astrophysics, 380(2):776–788, 2001.
  43. Physics-informed neural network for ultrasound nondestructive quantification of surface breaking cracks. Journal of Nondestructive Evaluation, 39(3):1–20, 2020.
  44. I. M. Sobol’. On the distribution of points in a cube and the approximate evaluation of integrals. Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 7(4):784–802, 1967.
  45. D. Stamatellos and A. P. Whitworth. Probing the initial conditions for star formation with monte carlo radiative transfer simulations. In Numerical Methods in Multidimensional Radiative Transfer, G. Kanschat, E. Meinköhn, R. Rannacher, and R. Wehrse, eds, pages 289–298. Springer, 2008.
  46. J. Stoer and R. Bulirsch. Introduction to numerical analysis. Springer Verlag, 2002.
  47. G. Widmer. Sparse Finite Elements for Radiative Transfer. PhD thesis, ETH Zurich, 2009.
  48. Castro: A new compressible astrophysical solver. iii. multigroup radiation hydrodynamics. Astrophysical Journal (supplement series), 204(7):27 pp, 2013.
User Edit Pencil Streamline Icon: https://streamlinehq.com
Authors (2)
  1. Siddhartha Mishra (76 papers)
  2. Roberto Molinaro (36 papers)
Citations (93)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now