doi: 10.3934/krm.2021046
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Lagrangian dual framework for conservative neural network solutions of kinetic equations

1. 

Department of Mathematics, Pohang University of Science and Technology, Pohang, Republic of Korea

2. 

Stochastic Analysis and Application Research Center, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea

* Corresponding author: Hyung Ju Hwang

Received  June 2021 Early access December 2021

In this paper, we propose a novel conservative formulation for solving kinetic equations via neural networks. More precisely, we formulate the learning problem as a constrained optimization problem with constraints that represent the physical conservation laws. The constraints are relaxed toward the residual loss function by the Lagrangian duality. By imposing physical conservation properties of the solution as constraints of the learning problem, we demonstrate far more accurate approximations of the solutions in terms of errors and the conservation laws, for the kinetic Fokker-Planck equation and the homogeneous Boltzmann equation.

Citation: Hyung Ju Hwang, Hwijae Son. Lagrangian dual framework for conservative neural network solutions of kinetic equations. Kinetic & Related Models, doi: 10.3934/krm.2021046
References:
[1]

V. V. Aristov, Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows, Fluid Mechanics and its Applications, 60, Kluwer Academic Publishers Group, Dordrecht, 2001. doi: 10.1007/978-94-010-0866-2.  Google Scholar

[2]

J. Berg and K. Nyström, A unified deep artificial neural network approach to partial differential equations in complex geometries, Neurocomputing, 317 (2018), 28-41.  doi: 10.1016/j.neucom.2018.06.056.  Google Scholar

[3]

D. P. Bertsekas, Multiplier methods: A survey, Automatica J. IFAC, 12 (1976), 133-145.  doi: 10.1016/0005-1098(76)90077-7.  Google Scholar

[4]

A. V. Bobylëv, Exact solutions of the Boltzmann equation, Dokl. Akad. Nauk SSSR, 225 (1975), 1296-1299.   Google Scholar

[5]

L. L. BonillaJ. A. Carrillo and J. Soler, Asymptotic behavior of an initial-boundary value problem for the Vlasov–Poisson–Fokker–Planck system, SIAM J. Appl. Math., 57 (1997), 1343-1372.  doi: 10.1137/S0036139995291544.  Google Scholar

[6]

G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control Signals Systems, 2 (1989), 303-314.  doi: 10.1007/BF02551274.  Google Scholar

[7]

G. DimarcoR. LoubèreJ. Narski and T. Rey, An efficient numerical method for solving the Boltzmann equation in multidimensions, J. Comput. Phys., 353 (2018), 46-81.  doi: 10.1016/j.jcp.2017.10.010.  Google Scholar

[8]

G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numer., 23 (2014), 369-520.  doi: 10.1017/S0962492914000063.  Google Scholar

[9]

W. EJ. Han and A. Jentzen, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Commun. Math. Stat., 5 (2017), 349-380.  doi: 10.1007/s40304-017-0117-6.  Google Scholar

[10]

W. E and B. Yu, The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems, Commun. Math. Stat., 6 (2018), 1-12.  doi: 10.1007/s40304-018-0127-z.  Google Scholar

[11]

F. Filbet and G. Russo, Accurate numerical methods for the Boltzmann equation, in Modeling and Computational Methods for Kinetic Equations, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Boston, MA, 2004,117–145.  Google Scholar

[12]

F. Fioretto, P. Van Hentenryck, T. W. K. Mak, C. Tran, F. Baldo and M. Lombardi, Lagrangian duality for constrained deep learning, in Machine Learning and Knowledge Discovery in Databases. Applied Data Science and Demo Track, Lecture Notes in Computer Science, 12461, Springer, Cham, 118–135. doi: 10.1007/978-3-030-67670-4_8.  Google Scholar

[13]

J. HanA. Jentzen and W. E, Solving high-dimensional partial differential equations using deep learning, Proc. Natl. Acad. Sci. USA, 115 (2018), 8505-8510.  doi: 10.1073/pnas.1718942115.  Google Scholar

[14]

K. HornikM. Stinchcombe and H. White, Multilayer feedforward networks are universal approximators, Neural Networks, 2 (1989), 359-366.  doi: 10.1016/0893-6080(89)90020-8.  Google Scholar

[15]

H. J. Hwang, J. W. Jang, H. Jo and J. Y. Lee, Trend to equilibrium for the kinetic Fokker-Planck equation via the neural network approach, J. Comput. Phys., 419 (2020), 25pp. doi: 10.1016/j.jcp.2020.109665.  Google Scholar

[16]

H. JoH. SonH. J. Hwang and E. H. Kim, Deep neural network approach to forward-inverse problems, Netw. Heterog. Media, 15 (2020), 247-259.  doi: 10.3934/nhm.2020011.  Google Scholar

[17]

E. Kharazmi, Z. Zhang and G. E. M. Karniadakis, hp-VPINNs: Variational physics-informed neural networks with domain decomposition, Comput. Methods Appl. Mech. Engrg., 374 (2021), 25pp. doi: 10.1016/j.cma.2020.113547.  Google Scholar

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D. P. Kingma and J. L. Ba, Adam: A method for stochastic optimization, preprint, arXiv: 1412.6980. Google Scholar

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M. Krook and T. T. Wu, Exact solutions of the Boltzmann equation, Phys. Fluids, 20 (1977), 1589-1595.  doi: 10.1063/1.861780.  Google Scholar

[20]

I. E. LagarisA. Likas and D. I. Fotiadis, Artificial neural networks for solving ordinary and partial differential equations, IEEE Trans. Neural Networks, 9 (1998), 987-1000.  doi: 10.1109/72.712178.  Google Scholar

[21]

I. E. LagarisA. C. Likas and D. G. Papageorgiou, Neural-network methods for boundary value problems with irregular boundaries, IEEE Trans. Neural Networks, 11 (2000), 1041-1049.  doi: 10.1109/72.870037.  Google Scholar

[22]

J. Y. LeeJ. W. Jang and H. J. Hwang, The model reduction of the Vlasov-Poisson-Fokker-Planck system to the Poisson-Nernst-Planck system via the deep neural network approach, ESAIM Math. Model. Numer. Anal., 55 (2021), 1803-1846.  doi: 10.1051/m2an/2021038.  Google Scholar

[23]

M. LeshnoV. Y. LinA. Pinkus and S. Schocken, Multilayer feedforward networks with a nonpolynomial activation function can approximate any function, Neural Networks, 6 (1993), 861-867.  doi: 10.1016/S0893-6080(05)80131-5.  Google Scholar

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X. Li, Simultaneous approximations of multivariate functions and their derivatives by neural networks with one hidden layer, Neurocomputing, 12 (1996), 327-343.  doi: 10.1016/0925-2312(95)00070-4.  Google Scholar

[25]

Y. Liao and P. Ming, Deep Nitsche method: Deep Ritz method with essential boundary conditions, Commun. Comput. Phys., 29 (2021), 1365-1384.  doi: 10.4208/cicp.OA-2020-0219.  Google Scholar

[26]

Q. Lou, X. Meng and G. E. Karniadakis, Physics-informed neural networks for solving forward and inverse flow problems via the Boltzmann-BGK formulation, J. Comput. Phys., 447 (2021), 20pp. doi: 10.1016/j.jcp.2021.110676.  Google Scholar

[27]

L. LuX. MengZ. Mao and G. E. Karniadakis, DeepXDE: A deep learning library for solving differential equations, SIAM Rev., 63 (2021), 208-228.  doi: 10.1137/19M1274067.  Google Scholar

[28]

D. G. Luenberger, Introduction to Linear and Nonlinear Programming, Vol. 28, Addison-Wesley, Reading, MA, 1973. Google Scholar

[29]

L. LyuK. WuR. Du and J. Chen, Enforcing exact boundary and initial conditions in the deep mixed residual method, CSIAM Trans. Appl. Math., 2 (2021), 748-775.  doi: 10.4208/csiam-am.SO-2021-0011.  Google Scholar

[30]

P. Márquez-Neila, M. Salzmann and P. Fua, Imposing hard constraints on deep networks: Promises and limitations, preprint, arXiv: 1706.02025. Google Scholar

[31]

L. D. McClenny and U. Braga-Neto, Self-adaptive physics-informed neural networks using a soft attention mechanism, preprint, arXiv: 2009.04544. Google Scholar

[32]

J. Müller and M. Zeinhofer, Deep Ritz revisited, preprint, arXiv: 1912.03937. Google Scholar

[33]

J. Müller and M. Zeinhofer, Notes on exact boundary values in residual minimisation, preprint, arXiv: 2105.02550. Google Scholar

[34]

Y. Nandwani, A. Pathak and P. Singla, A primal dual formulation for deep learning with constraints., Available from: https://proceedings.neurips.cc/paper/2019/file/cf708fc1decf0337aded484f8f4519ae-Paper.pdf. Google Scholar

[35]

L. Pareschi and G. Russo, Numerical solution of the Boltzmann equation. I. Spectrally accurate approximation of the collision operator, SIAM J. Numer. Anal., 37 (2000), 1217-1245.  doi: 10.1137/S0036142998343300.  Google Scholar

[36]

A. Paszke, S. Gross, F. Massa, A. Lerer and J. Bradbury, et al., PyTorch: An imperative style, high-performance deep learning library, in Advances in Neural Information Processing Systems, 2019, 8024–8035. Available from: https://proceedings.neurips.cc/paper/2019/file/bdbca288fee7f92f2bfa9f7012727740-Paper.pdf. Google Scholar

[37]

M. RaissiP. 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 (2019), 686-707.  doi: 10.1016/j.jcp.2018.10.045.  Google Scholar

[38]

S. N. RaviT. DinhV. S. Lokhande and V. Singh, Explicitly imposing constraints in deep networks via conditional gradients gives improved generalization and faster convergence, Proceedings of the AAAI Conference on Artificial Intelligence, 33 (2019), 4772-4779.  doi: 10.1609/aaai.v33i01.33014772.  Google Scholar

[39]

S. Sangalli, E. Erdil, A. Hoetker, O. Donati and E. Konukoglu, Constrained optimization to train neural networks on critical and under-represented classes, preprint, arXiv: 2102.12894. Google Scholar

[40]

J. Sirignano and K. Spiliopoulos, DGM: A deep learning algorithm for solving partial differential equations, J. Comput. Phys., 375 (2018), 1339-1364.  doi: 10.1016/j.jcp.2018.08.029.  Google Scholar

[41]

H. Son, J. W. Jang, W. J. Han and H. J. Hwang, Sobolev training for physics informed neural networks, preprint, arXiv: 2101.08932. Google Scholar

[42]

R. van der Meer, C. W. Oosterlee and A. Borovykh, Optimally weighted loss functions for solving PDEs with Neural Networks, J. Comput. Appl. Math., 405 (2022). doi: 10.1016/j.cam.2021.113887.  Google Scholar

[43]

S. Wang, X. Yu and P. Perdikaris, When and why PINNs fail to train: A neural tangent kernel perspective, J. Comput. Phys., 449 (2022). doi: 10.1016/j.jcp.2021.110768.  Google Scholar

[44]

S. Wollman and E. Ozizmir, A deterministic particle method for the Vlasov–Fokker–Planck equation in one dimension, J. Comput. Appl. Math., 213 (2008), 316-365.  doi: 10.1016/j.cam.2007.01.008.  Google Scholar

show all references

References:
[1]

V. V. Aristov, Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows, Fluid Mechanics and its Applications, 60, Kluwer Academic Publishers Group, Dordrecht, 2001. doi: 10.1007/978-94-010-0866-2.  Google Scholar

[2]

J. Berg and K. Nyström, A unified deep artificial neural network approach to partial differential equations in complex geometries, Neurocomputing, 317 (2018), 28-41.  doi: 10.1016/j.neucom.2018.06.056.  Google Scholar

[3]

D. P. Bertsekas, Multiplier methods: A survey, Automatica J. IFAC, 12 (1976), 133-145.  doi: 10.1016/0005-1098(76)90077-7.  Google Scholar

[4]

A. V. Bobylëv, Exact solutions of the Boltzmann equation, Dokl. Akad. Nauk SSSR, 225 (1975), 1296-1299.   Google Scholar

[5]

L. L. BonillaJ. A. Carrillo and J. Soler, Asymptotic behavior of an initial-boundary value problem for the Vlasov–Poisson–Fokker–Planck system, SIAM J. Appl. Math., 57 (1997), 1343-1372.  doi: 10.1137/S0036139995291544.  Google Scholar

[6]

G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control Signals Systems, 2 (1989), 303-314.  doi: 10.1007/BF02551274.  Google Scholar

[7]

G. DimarcoR. LoubèreJ. Narski and T. Rey, An efficient numerical method for solving the Boltzmann equation in multidimensions, J. Comput. Phys., 353 (2018), 46-81.  doi: 10.1016/j.jcp.2017.10.010.  Google Scholar

[8]

G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numer., 23 (2014), 369-520.  doi: 10.1017/S0962492914000063.  Google Scholar

[9]

W. EJ. Han and A. Jentzen, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Commun. Math. Stat., 5 (2017), 349-380.  doi: 10.1007/s40304-017-0117-6.  Google Scholar

[10]

W. E and B. Yu, The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems, Commun. Math. Stat., 6 (2018), 1-12.  doi: 10.1007/s40304-018-0127-z.  Google Scholar

[11]

F. Filbet and G. Russo, Accurate numerical methods for the Boltzmann equation, in Modeling and Computational Methods for Kinetic Equations, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Boston, MA, 2004,117–145.  Google Scholar

[12]

F. Fioretto, P. Van Hentenryck, T. W. K. Mak, C. Tran, F. Baldo and M. Lombardi, Lagrangian duality for constrained deep learning, in Machine Learning and Knowledge Discovery in Databases. Applied Data Science and Demo Track, Lecture Notes in Computer Science, 12461, Springer, Cham, 118–135. doi: 10.1007/978-3-030-67670-4_8.  Google Scholar

[13]

J. HanA. Jentzen and W. E, Solving high-dimensional partial differential equations using deep learning, Proc. Natl. Acad. Sci. USA, 115 (2018), 8505-8510.  doi: 10.1073/pnas.1718942115.  Google Scholar

[14]

K. HornikM. Stinchcombe and H. White, Multilayer feedforward networks are universal approximators, Neural Networks, 2 (1989), 359-366.  doi: 10.1016/0893-6080(89)90020-8.  Google Scholar

[15]

H. J. Hwang, J. W. Jang, H. Jo and J. Y. Lee, Trend to equilibrium for the kinetic Fokker-Planck equation via the neural network approach, J. Comput. Phys., 419 (2020), 25pp. doi: 10.1016/j.jcp.2020.109665.  Google Scholar

[16]

H. JoH. SonH. J. Hwang and E. H. Kim, Deep neural network approach to forward-inverse problems, Netw. Heterog. Media, 15 (2020), 247-259.  doi: 10.3934/nhm.2020011.  Google Scholar

[17]

E. Kharazmi, Z. Zhang and G. E. M. Karniadakis, hp-VPINNs: Variational physics-informed neural networks with domain decomposition, Comput. Methods Appl. Mech. Engrg., 374 (2021), 25pp. doi: 10.1016/j.cma.2020.113547.  Google Scholar

[18]

D. P. Kingma and J. L. Ba, Adam: A method for stochastic optimization, preprint, arXiv: 1412.6980. Google Scholar

[19]

M. Krook and T. T. Wu, Exact solutions of the Boltzmann equation, Phys. Fluids, 20 (1977), 1589-1595.  doi: 10.1063/1.861780.  Google Scholar

[20]

I. E. LagarisA. Likas and D. I. Fotiadis, Artificial neural networks for solving ordinary and partial differential equations, IEEE Trans. Neural Networks, 9 (1998), 987-1000.  doi: 10.1109/72.712178.  Google Scholar

[21]

I. E. LagarisA. C. Likas and D. G. Papageorgiou, Neural-network methods for boundary value problems with irregular boundaries, IEEE Trans. Neural Networks, 11 (2000), 1041-1049.  doi: 10.1109/72.870037.  Google Scholar

[22]

J. Y. LeeJ. W. Jang and H. J. Hwang, The model reduction of the Vlasov-Poisson-Fokker-Planck system to the Poisson-Nernst-Planck system via the deep neural network approach, ESAIM Math. Model. Numer. Anal., 55 (2021), 1803-1846.  doi: 10.1051/m2an/2021038.  Google Scholar

[23]

M. LeshnoV. Y. LinA. Pinkus and S. Schocken, Multilayer feedforward networks with a nonpolynomial activation function can approximate any function, Neural Networks, 6 (1993), 861-867.  doi: 10.1016/S0893-6080(05)80131-5.  Google Scholar

[24]

X. Li, Simultaneous approximations of multivariate functions and their derivatives by neural networks with one hidden layer, Neurocomputing, 12 (1996), 327-343.  doi: 10.1016/0925-2312(95)00070-4.  Google Scholar

[25]

Y. Liao and P. Ming, Deep Nitsche method: Deep Ritz method with essential boundary conditions, Commun. Comput. Phys., 29 (2021), 1365-1384.  doi: 10.4208/cicp.OA-2020-0219.  Google Scholar

[26]

Q. Lou, X. Meng and G. E. Karniadakis, Physics-informed neural networks for solving forward and inverse flow problems via the Boltzmann-BGK formulation, J. Comput. Phys., 447 (2021), 20pp. doi: 10.1016/j.jcp.2021.110676.  Google Scholar

[27]

L. LuX. MengZ. Mao and G. E. Karniadakis, DeepXDE: A deep learning library for solving differential equations, SIAM Rev., 63 (2021), 208-228.  doi: 10.1137/19M1274067.  Google Scholar

[28]

D. G. Luenberger, Introduction to Linear and Nonlinear Programming, Vol. 28, Addison-Wesley, Reading, MA, 1973. Google Scholar

[29]

L. LyuK. WuR. Du and J. Chen, Enforcing exact boundary and initial conditions in the deep mixed residual method, CSIAM Trans. Appl. Math., 2 (2021), 748-775.  doi: 10.4208/csiam-am.SO-2021-0011.  Google Scholar

[30]

P. Márquez-Neila, M. Salzmann and P. Fua, Imposing hard constraints on deep networks: Promises and limitations, preprint, arXiv: 1706.02025. Google Scholar

[31]

L. D. McClenny and U. Braga-Neto, Self-adaptive physics-informed neural networks using a soft attention mechanism, preprint, arXiv: 2009.04544. Google Scholar

[32]

J. Müller and M. Zeinhofer, Deep Ritz revisited, preprint, arXiv: 1912.03937. Google Scholar

[33]

J. Müller and M. Zeinhofer, Notes on exact boundary values in residual minimisation, preprint, arXiv: 2105.02550. Google Scholar

[34]

Y. Nandwani, A. Pathak and P. Singla, A primal dual formulation for deep learning with constraints., Available from: https://proceedings.neurips.cc/paper/2019/file/cf708fc1decf0337aded484f8f4519ae-Paper.pdf. Google Scholar

[35]

L. Pareschi and G. Russo, Numerical solution of the Boltzmann equation. I. Spectrally accurate approximation of the collision operator, SIAM J. Numer. Anal., 37 (2000), 1217-1245.  doi: 10.1137/S0036142998343300.  Google Scholar

[36]

A. Paszke, S. Gross, F. Massa, A. Lerer and J. Bradbury, et al., PyTorch: An imperative style, high-performance deep learning library, in Advances in Neural Information Processing Systems, 2019, 8024–8035. Available from: https://proceedings.neurips.cc/paper/2019/file/bdbca288fee7f92f2bfa9f7012727740-Paper.pdf. Google Scholar

[37]

M. RaissiP. 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 (2019), 686-707.  doi: 10.1016/j.jcp.2018.10.045.  Google Scholar

[38]

S. N. RaviT. DinhV. S. Lokhande and V. Singh, Explicitly imposing constraints in deep networks via conditional gradients gives improved generalization and faster convergence, Proceedings of the AAAI Conference on Artificial Intelligence, 33 (2019), 4772-4779.  doi: 10.1609/aaai.v33i01.33014772.  Google Scholar

[39]

S. Sangalli, E. Erdil, A. Hoetker, O. Donati and E. Konukoglu, Constrained optimization to train neural networks on critical and under-represented classes, preprint, arXiv: 2102.12894. Google Scholar

[40]

J. Sirignano and K. Spiliopoulos, DGM: A deep learning algorithm for solving partial differential equations, J. Comput. Phys., 375 (2018), 1339-1364.  doi: 10.1016/j.jcp.2018.08.029.  Google Scholar

[41]

H. Son, J. W. Jang, W. J. Han and H. J. Hwang, Sobolev training for physics informed neural networks, preprint, arXiv: 2101.08932. Google Scholar

[42]

R. van der Meer, C. W. Oosterlee and A. Borovykh, Optimally weighted loss functions for solving PDEs with Neural Networks, J. Comput. Appl. Math., 405 (2022). doi: 10.1016/j.cam.2021.113887.  Google Scholar

[43]

S. Wang, X. Yu and P. Perdikaris, When and why PINNs fail to train: A neural tangent kernel perspective, J. Comput. Phys., 449 (2022). doi: 10.1016/j.jcp.2021.110768.  Google Scholar

[44]

S. Wollman and E. Ozizmir, A deterministic particle method for the Vlasov–Fokker–Planck equation in one dimension, J. Comput. Appl. Math., 213 (2008), 316-365.  doi: 10.1016/j.cam.2007.01.008.  Google Scholar

Figure 1.  Left: Relaxation to Maxwellian of f(t, x = 1, v) from an initial condition $ f_0^{(1)} $. Right: Relaxation to Maxwellian of f(t, x = 1, v) from an initial condition $ f_0^{(2)} $
Figure 2.  Numerical solutions of Test 1 at $ t = 0, \frac{1}{2}, $ and $ 1 $
Figure 3.  Left: Loss value in training epoch at log scale. Right: Actual error in training epoch at log scale
Figure 4.  Left: Time averaged mass in training epoch. Right: Total mass in time after training is finished
Figure 5.  Numerical solutions of Test 1 at $ t = 0, \frac{3}{2}, $ and $ 3 $
Figure 6.  Left: Loss value in training epoch at log scale. Right: Actual error in training epoch at log scale
Figure 7.  Left: Time averaged mass in training epoch. Right: Total mass in time after training is finished
Figure 8.  Left: Value of the loss $ \widehat{Loss}_B $ in training epoch. Right: $ L^{\infty}((0,1);L^2_{v_x,v_y}) $ error in training epoch
Figure 9.  Left: Total mass in time after the training is finished. Right: Kinetic energy in time after the training is finished
Figure 10.  Momentum in time after the training is finished
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