Computing Lyapunov functions using deep neural networks

Lars Grüne

doi:10.3934/jcd.2021006

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April 2021, 8(2): 131-152. doi: 10.3934/jcd.2021006

Computing Lyapunov functions using deep neural networks

Lars Grüne

Mathematical Institute, University of Bayreuth, 95440 Bayreuth, Germany

Received September 2020 Revised November 2020 Published April 2021 Early access December 2020

We propose a deep neural network architecture and associated loss functions for a training algorithm for computing approximate Lyapunov functions of systems of nonlinear ordinary differential equations. Under the assumption that the system admits a compositional Lyapunov function, we prove that the number of neurons needed for an approximation of a Lyapunov function with fixed accuracy grows only polynomially in the state dimension, i.e., the proposed approach is able to overcome the curse of dimensionality. We show that nonlinear systems satisfying a small-gain condition admit compositional Lyapunov functions. Numerical examples in up to ten space dimensions illustrate the performance of the training scheme.

Keywords: Deep neural network, Lyapunov function, stability, small-gain condition, curse of dimensionality, training algorithm.

Mathematics Subject Classification: Primary:68T07;34D20;Secondary:93D09.

Citation: Lars Grüne. Computing Lyapunov functions using deep neural networks. Journal of Computational Dynamics, 2021, 8 (2) : 131-152. doi: 10.3934/jcd.2021006

References:

[1]	M. Abadi, A. Agarwal, P. Barham, E. Brevdo and Z. Chen, et al., TensorFlow: Large-scale machine learning on heterogeneous systems, 2015., Available from: https://www.tensorflow.org/.
[2]	M. Abu-Khalaf and F. L. Lewis, Nearly optimal control laws for nonlinear systems with saturating actuators using a neural network HJB approach, Automatica J. IFAC, 41 (2005), 779-791. doi: 10.1016/j.automatica.2004.11.034.
[3]	J. Anderson and A. Papachristodoulou, Advances in computational Lyapunov analysis using sum-of-squares programming, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2361-2381. doi: 10.3934/dcdsb.2015.20.2361.
[4]	J. Berner, P. Grohs and A. Jentzen, Analysis of the generalization error: Empirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations, SIAM J. Math. Data Sci., 2 (2020), 631-657. doi: 10.1137/19M125649X.
[5]	L. Bottou, Large-scale machine learning with stochastic gradient descent, in Proceedings of COMPSTAT'2010, Physica-Verlag/Springer, Heidelberg, 2010, 177-186. doi: 10.1007/978-3-7908-2604-3_16.
[6]	L. Bottou, F. E. Curtis and J. Nocedal, Optimization methods for large-scale machine learning, SIAM Rev., 60 (2018), 223-311. doi: 10.1137/16M1080173.
[7]	F. Camilli, L. Grüne and F. Wirth, A regularization of Zubov's equation for robust domains of attraction, in Nonlinear Control in the Year 2000, Lect. Notes Control Inf. Sci., 258, NCN, Springer, London, 2001, 277-289. doi: 10.1007/BFb0110220.
[8]	F. Camilli, L. Grüne and F. Wirth, Domains of attraction of interconnected systems: A Zubov method approach, European Control Conference (ECC), Budapest, Hungary, 2009. doi: 10.23919/ECC.2009.7074385.
[9]	G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control Signals Systems, 2 (1989), 303-314. doi: 10.1007/BF02551274.
[10]	J. Darbon, G. P. Langlois and T. Meng, Overcoming the curse of dimensionality for some Hamilton-Jacobi partial differential equations via neural network architectures, Res. Math. Sci., 7 (2020), 50pp. doi: 10.1007/s40687-020-00215-6.
[11]	S. Dashkovskiy, H. Ito and F. Wirth, On a small gain theorem for ISS networks in dissipative Lyapunov form, Eur. J. Control, 17 (2011), 357-365. doi: 10.3166/ejc.17.357-365.
[12]	S. N. Dashkovskiy, B. S. Rüffer and F. R. Wirth, Small gain theorems for large scale systems and construction of ISS Lyapunov functions, SIAM J. Control Optim., 48 (2010), 4089-4118. doi: 10.1137/090746483.
[13]	W. E, J. 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.
[14]	P. Giesl and S. Hafstein, Computation of Lyapunov functions for nonlinear discrete time systems by linear programming, J. Difference Equ. Appl., 20 (2014), 610-640. doi: 10.1080/10236198.2013.867341.
[15]	P. Giesl and S. Hafstein, Review on computational methods for Lyapunov functions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2291-2331. doi: 10.3934/dcdsb.2015.20.2291.
[16]	P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Mathematics, 1904, Springer, Berlin, 2007. doi: 10.1007/978-3-540-69909-5.
[17]	L. Grüne, Overcoming the curse of dimensionality for approximating Lyapunov functions with deep neural networks under a small-gain condition, preprint, arXiv: 2001.08423.
[18]	S. Hafstein, C. M. Kellett and H. Li, Continuous and piecewise affine Lyapunov functions using the Yoshizawa construction, American Control Conference, Portland, OR, 2014. doi: 10.1109/ACC.2014.6858660.
[19]	S. F. Hafstein, An algorithm for constructing Lyapunov functions, Electronic Journal of Differential Equations, Monograph, 8, Texas State University-San Marcos, Department of Mathematics, San Marcos, TX, 2007, 100pp.
[20]	W. Hahn, Stability of Motion, Die Grundlehren der mathematischen Wissenschaften, 138, Springer-Verlag New York, Inc., New York, 1967. doi: 10.1007/978-3-642-50085-5.
[21]	J. Han, A. 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.
[22]	K. Hornik, M. Stinchcombe and H. White, Multilayer feedforward networks are universal approximators, Neural Networks, 2 (1989), 359-366. doi: 10.1016/0893-6080(89)90020-8.
[23]	C. Huré, H. Pham and X. Warin, Deep backward schemes for high-dimensional nonlinear PDEs, Math. Comp., 89 (2020), 1547-1579. doi: 10.1090/mcom/3514.
[24]	M. Hutzenthaler, A. Jentzen and T. Kruse, Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities, preprint, arXiv: 1912.02571.
[25]	M. Hutzenthaler, A. Jentzen, T. Kruse and T. A. Nguyen, A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations, SN Partial Differ. Equ. Appl., 10 (2020). doi: 10.1007/s42985-019-0006-9.
[26]	Z.-P. Jiang, A. R. Teel and L. Praly, Small-gain theorem for ISS systems and applications, Math. Control Signals Systems, 7 (1994), 95-120. doi: 10.1007/BF01211469.
[27]	Z.-P. Jiang, I. M. Y. Mareels and Y. Wang, A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems, Automatica J. IFAC, 32 (1996), 1211-1215. doi: 10.1016/0005-1098(96)00051-9.
[28]	H. K. Khalil, Nonlinear Systems, Prentice-Hall, 1996.
[29]	S. Mohammad Khansari-Zadeh and A. Billard, Learning control Lyapunov function to ensure stability of dynamical system-based robot reaching motions, Robotics and Autonomous Systems, 62 (2014), 752-765. doi: 10.1016/j.robot.2014.03.001.
[30]	N. E. Kirin, R. A. Nelepin and V. N. Ba${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$daev, Construction of the attraction region by Zubov's method, Differ. Equations, 17 (1982), 871-880.
[31]	F. L. Lewis, S. Jagannathan and A. Yeşildirek, Neural Network Control of Robot Manipulators and Nonlinear Systems, Taylor and Francis, 1998.
[32]	H. Li, Computation of Lyapunov Functions and Stability of Interconnected Systems, Ph.D dissertation, Universität Bayreuth, 2015.
[33]	Y. Long and M. M. Bayoumi, Feedback stabilization: Control Lyapunov functions modelled by neural networks, Proceedings of the 32nd IEEE Conference on Decision and Control, San Antonio, TX, 1993. doi: 10.1109/CDC.1993.325708.
[34]	H. N. Mhaskar, Neural networks for optimal approximation of smooth and analytic functions, Neural Comput., 8 (1996), 164-177. doi: 10.1162/neco.1996.8.1.164.
[35]	N. Noroozi, P. Karimaghaee, F. Safaei and H. Javadi, Generation of Lyapunov functions by neural networks, Proceedings of the World Congress on Engineering. Vol I, London, UK, 2008.
[36]	V. Petridis and S. Petridis, Construction of neural network based Lyapunov functions, Proceedings of the International Joint Conference on Neural Networks, Vancouver, Canada, 2006, 5059-5065.
[37]	T. Poggio, H. Mhaskar, L. Rosaco, B. Miranda and Q. Liao, Why and when can deep - but not shallow - networks avoid the curse of dimensionality: A review, Int. J. Automat. Computing, 14 (2017), 503-519. doi: 10.1007/s11633-017-1054-2.
[38]	C. Reisinger and Y. Zhang, Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems, Anal. Appl. (Singap.), 18 (2020), 951--999. doi: 10.1142/S0219530520500116.
[39]	S. M. Richards, F. Berkenkamp and A. Krause, The Lyapunov neural network: Adaptive stability certification for safe learning of dynamical systems, Proceedings of the 2nd Conference on Robot Learning - CoRL 2018, Zürich, Switzerland, 2018. Available from: arXiv: 1808.00924.
[40]	B. S. Rüffer, Monotone Systems, Graphs, and Stability of Large-Scale Interconnected Systems, Ph.D dissertation, Universität Bremen, Germany, 2007.
[41]	G. Serpen, Empirical approximation for Lyapunov functions with artificial neural nets, Proc. International Joint Conference on Neural Networks, Montreal, Que., Canada, 2005. doi: 10.1109/IJCNN.2005.1555943.
[42]	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.
[43]	E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443. doi: 10.1109/9.28018.
[44]	E. D. Sontag, Feedback stabilization using two-hidden-layer nets, IEEE Trans. Neural Networks, 3 (1992), 981-990. doi: 10.1109/72.165599.
[45]	V. I. Zubov, Methods of A.M. Lyapunov and Their Application, P. Noordhoff Ltd, Groningen, 1964.

show all references

References:

[1]	M. Abadi, A. Agarwal, P. Barham, E. Brevdo and Z. Chen, et al., TensorFlow: Large-scale machine learning on heterogeneous systems, 2015., Available from: https://www.tensorflow.org/.
[2]	M. Abu-Khalaf and F. L. Lewis, Nearly optimal control laws for nonlinear systems with saturating actuators using a neural network HJB approach, Automatica J. IFAC, 41 (2005), 779-791. doi: 10.1016/j.automatica.2004.11.034.
[3]	J. Anderson and A. Papachristodoulou, Advances in computational Lyapunov analysis using sum-of-squares programming, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2361-2381. doi: 10.3934/dcdsb.2015.20.2361.
[4]	J. Berner, P. Grohs and A. Jentzen, Analysis of the generalization error: Empirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations, SIAM J. Math. Data Sci., 2 (2020), 631-657. doi: 10.1137/19M125649X.
[5]	L. Bottou, Large-scale machine learning with stochastic gradient descent, in Proceedings of COMPSTAT'2010, Physica-Verlag/Springer, Heidelberg, 2010, 177-186. doi: 10.1007/978-3-7908-2604-3_16.
[6]	L. Bottou, F. E. Curtis and J. Nocedal, Optimization methods for large-scale machine learning, SIAM Rev., 60 (2018), 223-311. doi: 10.1137/16M1080173.
[7]	F. Camilli, L. Grüne and F. Wirth, A regularization of Zubov's equation for robust domains of attraction, in Nonlinear Control in the Year 2000, Lect. Notes Control Inf. Sci., 258, NCN, Springer, London, 2001, 277-289. doi: 10.1007/BFb0110220.
[8]	F. Camilli, L. Grüne and F. Wirth, Domains of attraction of interconnected systems: A Zubov method approach, European Control Conference (ECC), Budapest, Hungary, 2009. doi: 10.23919/ECC.2009.7074385.
[9]	G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control Signals Systems, 2 (1989), 303-314. doi: 10.1007/BF02551274.
[10]	J. Darbon, G. P. Langlois and T. Meng, Overcoming the curse of dimensionality for some Hamilton-Jacobi partial differential equations via neural network architectures, Res. Math. Sci., 7 (2020), 50pp. doi: 10.1007/s40687-020-00215-6.
[11]	S. Dashkovskiy, H. Ito and F. Wirth, On a small gain theorem for ISS networks in dissipative Lyapunov form, Eur. J. Control, 17 (2011), 357-365. doi: 10.3166/ejc.17.357-365.
[12]	S. N. Dashkovskiy, B. S. Rüffer and F. R. Wirth, Small gain theorems for large scale systems and construction of ISS Lyapunov functions, SIAM J. Control Optim., 48 (2010), 4089-4118. doi: 10.1137/090746483.
[13]	W. E, J. 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.
[14]	P. Giesl and S. Hafstein, Computation of Lyapunov functions for nonlinear discrete time systems by linear programming, J. Difference Equ. Appl., 20 (2014), 610-640. doi: 10.1080/10236198.2013.867341.
[15]	P. Giesl and S. Hafstein, Review on computational methods for Lyapunov functions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2291-2331. doi: 10.3934/dcdsb.2015.20.2291.
[16]	P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Mathematics, 1904, Springer, Berlin, 2007. doi: 10.1007/978-3-540-69909-5.
[17]	L. Grüne, Overcoming the curse of dimensionality for approximating Lyapunov functions with deep neural networks under a small-gain condition, preprint, arXiv: 2001.08423.
[18]	S. Hafstein, C. M. Kellett and H. Li, Continuous and piecewise affine Lyapunov functions using the Yoshizawa construction, American Control Conference, Portland, OR, 2014. doi: 10.1109/ACC.2014.6858660.
[19]	S. F. Hafstein, An algorithm for constructing Lyapunov functions, Electronic Journal of Differential Equations, Monograph, 8, Texas State University-San Marcos, Department of Mathematics, San Marcos, TX, 2007, 100pp.
[20]	W. Hahn, Stability of Motion, Die Grundlehren der mathematischen Wissenschaften, 138, Springer-Verlag New York, Inc., New York, 1967. doi: 10.1007/978-3-642-50085-5.
[21]	J. Han, A. 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.
[22]	K. Hornik, M. Stinchcombe and H. White, Multilayer feedforward networks are universal approximators, Neural Networks, 2 (1989), 359-366. doi: 10.1016/0893-6080(89)90020-8.
[23]	C. Huré, H. Pham and X. Warin, Deep backward schemes for high-dimensional nonlinear PDEs, Math. Comp., 89 (2020), 1547-1579. doi: 10.1090/mcom/3514.
[24]	M. Hutzenthaler, A. Jentzen and T. Kruse, Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities, preprint, arXiv: 1912.02571.
[25]	M. Hutzenthaler, A. Jentzen, T. Kruse and T. A. Nguyen, A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations, SN Partial Differ. Equ. Appl., 10 (2020). doi: 10.1007/s42985-019-0006-9.
[26]	Z.-P. Jiang, A. R. Teel and L. Praly, Small-gain theorem for ISS systems and applications, Math. Control Signals Systems, 7 (1994), 95-120. doi: 10.1007/BF01211469.
[27]	Z.-P. Jiang, I. M. Y. Mareels and Y. Wang, A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems, Automatica J. IFAC, 32 (1996), 1211-1215. doi: 10.1016/0005-1098(96)00051-9.
[28]	H. K. Khalil, Nonlinear Systems, Prentice-Hall, 1996.
[29]	S. Mohammad Khansari-Zadeh and A. Billard, Learning control Lyapunov function to ensure stability of dynamical system-based robot reaching motions, Robotics and Autonomous Systems, 62 (2014), 752-765. doi: 10.1016/j.robot.2014.03.001.
[30]	N. E. Kirin, R. A. Nelepin and V. N. Ba${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$daev, Construction of the attraction region by Zubov's method, Differ. Equations, 17 (1982), 871-880.
[31]	F. L. Lewis, S. Jagannathan and A. Yeşildirek, Neural Network Control of Robot Manipulators and Nonlinear Systems, Taylor and Francis, 1998.
[32]	H. Li, Computation of Lyapunov Functions and Stability of Interconnected Systems, Ph.D dissertation, Universität Bayreuth, 2015.
[33]	Y. Long and M. M. Bayoumi, Feedback stabilization: Control Lyapunov functions modelled by neural networks, Proceedings of the 32nd IEEE Conference on Decision and Control, San Antonio, TX, 1993. doi: 10.1109/CDC.1993.325708.
[34]	H. N. Mhaskar, Neural networks for optimal approximation of smooth and analytic functions, Neural Comput., 8 (1996), 164-177. doi: 10.1162/neco.1996.8.1.164.
[35]	N. Noroozi, P. Karimaghaee, F. Safaei and H. Javadi, Generation of Lyapunov functions by neural networks, Proceedings of the World Congress on Engineering. Vol I, London, UK, 2008.
[36]	V. Petridis and S. Petridis, Construction of neural network based Lyapunov functions, Proceedings of the International Joint Conference on Neural Networks, Vancouver, Canada, 2006, 5059-5065.
[37]	T. Poggio, H. Mhaskar, L. Rosaco, B. Miranda and Q. Liao, Why and when can deep - but not shallow - networks avoid the curse of dimensionality: A review, Int. J. Automat. Computing, 14 (2017), 503-519. doi: 10.1007/s11633-017-1054-2.
[38]	C. Reisinger and Y. Zhang, Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems, Anal. Appl. (Singap.), 18 (2020), 951--999. doi: 10.1142/S0219530520500116.
[39]	S. M. Richards, F. Berkenkamp and A. Krause, The Lyapunov neural network: Adaptive stability certification for safe learning of dynamical systems, Proceedings of the 2nd Conference on Robot Learning - CoRL 2018, Zürich, Switzerland, 2018. Available from: arXiv: 1808.00924.
[40]	B. S. Rüffer, Monotone Systems, Graphs, and Stability of Large-Scale Interconnected Systems, Ph.D dissertation, Universität Bremen, Germany, 2007.
[41]	G. Serpen, Empirical approximation for Lyapunov functions with artificial neural nets, Proc. International Joint Conference on Neural Networks, Montreal, Que., Canada, 2005. doi: 10.1109/IJCNN.2005.1555943.
[42]	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.
[43]	E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443. doi: 10.1109/9.28018.
[44]	E. D. Sontag, Feedback stabilization using two-hidden-layer nets, IEEE Trans. Neural Networks, 3 (1992), 981-990. doi: 10.1109/72.165599.
[45]	V. I. Zubov, Methods of A.M. Lyapunov and Their Application, P. Noordhoff Ltd, Groningen, 1964.

Figure 1. Neural network with $ 1 $ and $ 2 $ hidden layers

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Figure 2. Neural network for Lyapunov functions, $ f\in F_1^{d_{\max}} $

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Figure 3. Neural network for Lyapunov functions, $ f\in F_2^{d_{\max}} $

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Figure 5. Attempt to compute a Lyapunov function $ W(\cdot;\theta^*) $ (solid) with its orbital derivative $ DW(\cdot;\theta^*)f $ (mesh) for Ex. (12) with loss function (9)

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Figure 4. Approximate Lyapunov function $ W(\cdot;\theta^*) $ (solid) and its orbital derivative $ DW(\cdot;\theta^*)f $ (mesh) for Example (12) computed with loss function (11)

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Figure 6. Approximate Lyapunov function $ W(\cdot;\theta^*) $ (solid) and its orbital derivative $ DW(\cdot;\theta^*)f $ (mesh) for Example (13) on $ (x_2,x_8) $-plane

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Figure 7. Approximate Lyapunov function $ W(\cdot;\theta^*) $ (solid) and its orbital derivative $ DW(\cdot;\theta^*)f $ (mesh) for Example (13) on $ (x_9,x_{10}) $-plane

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Figure 8. Value of approximate Lyapunov function $ W(x(t);\theta^*) $ along trajectories for initial values $ x_0 = (1,1,1,1,1,1,1,1,1,1)^T $, $ (0,1,0,1,0,1,0,1,0,1)^T $, $ (1,0,0,0,0,0,0,0,0,0)^T $ (left to right)

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