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Data-driven evolutions of critical points
1. | Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria |
2. | Department of Mathematics, Technical University of Munich, Boltzmannstrasse 3, 85748 Garching bei München, Germany |
3. | Institute of Mathematics and Scientific Computing, Karl-Franzens University of Graz, Heinrichstrasse 36/III, 8010 Graz, Austria |
In this paper we are concerned with the learnability of energies from data obtained by observing time evolutions of their critical points starting at random initial equilibria. As a byproduct of our theoretical framework we introduce the novel concept of mean-field limit of critical point evolutions and of their energy balance as a new form of transport. We formulate the energy learning as a variational problem, minimizing the discrepancy of energy competitors from fulfilling the equilibrium condition along any trajectory of critical points originated at random initial equilibria. By $ \Gamma $-convergence arguments we prove the convergence of minimal solutions obtained from finite number of observations to the exact energy in a suitable sense. The abstract framework is actually fully constructive and numerically implementable. Hence, the approximation of the energy from a finite number of observations of past evolutions allows one to simulate further evolutions, which are fully data-driven. As we aim at a precise quantitative analysis, and to provide concrete examples of tractable solutions, we present analytic and numerical results on the reconstruction of an elastic energy for a one-dimensional model of thin nonlinear-elastic rod.
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Singular vanishing-viscosity limits of gradient flows: The finite-dimensional case, J. Differential Equations, 263 (2017), 7815-7855.
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V. Agostiniani, R. Rossi and G. Savaré,
On the transversality conditions and their genericity, Rend. Circ. Mat. Palermo, 64 (2015), 101-116.
doi: 10.1007/s12215-014-0184-4. |
[3] |
V. Agostiniani, R. Rossi, and G. Savaré, Singular vanishing-viscosity limits of gradient flows in Hilbert spaces, personal communication: in preparation, (2018). Google Scholar |
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V. Albani, U. M. Ascher, X. Yang and J. P. Zubelli,
Data driven recovery of local volatility surfaces, Inverse Probl. Imaging, 11 (2017), 799-823.
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L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. |
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L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, second ed., 2008. |
[7] |
M. Bongini, M. Fornasier, M. Hansen and M. Maggioni,
Inferring interaction rules from observations of evolutive systems Ⅰ: The variational approach, Math. Models Methods Appl. Sci., 27 (2017), 909-951.
doi: 10.1142/S0218202517500208. |
[8] |
A. Braides, Γ-Convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications, 22. Oxford University Press, Oxford, 2002.
doi: 10.1093/acprof:oso/9780198507840.001.0001. |
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T. Q. Chen, Y. Rubanova, J. Bettencourt and D. K. Duvenaud, Neural ordinary differential equations, Advances in Neural Information Processing Systems, S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi and R. Garnett, eds., Curran Associates, Inc., 31 (2018), 6571–6583. Google Scholar |
[10] |
S. Conti, S. Müller and M. Ortiz,
Data-driven problems in elasticity, Arch. Ration. Mech. Anal., 229 (2018), 79-123.
doi: 10.1007/s00205-017-1214-0. |
[11] |
S. Crépey,
Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization, SIAM J. Math. Anal., 34 (2003), 1183-1206.
doi: 10.1137/S0036141001400202. |
[12] |
T. S. Cubitt, J. Eisert and M. M. Wolf, Extracting dynamical equations from experimental data is NP hard, Phys. Rev. Lett., 108 (2012), 120503.
doi: 10.1103/PhysRevLett.108.120503. |
[13] |
G. Dal Maso, An Introduction to $\Gamma$-Convergence, vol. 8 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc., Boston, MA, 1993.
doi: 10.1007/978-1-4612-0327-8. |
[14] |
W. E,
A proposal on machine learning via dynamical systems, Commun. Math. Stat., 5 (2017), 1-11.
doi: 10.1007/s40304-017-0103-z. |
[15] |
W. E, J. Han and Q. Li, A mean-field optimal control formulation of deep learning, Res. Math. Sci., 6 (2019), No. 10, 41 pp.
doi: 10.1007/s40687-018-0172-y. |
[16] |
H. Egger and H. W. Engl,
Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates, Inverse Problems, 21 (2005), 1027-1045.
doi: 10.1088/0266-5611/21/3/014. |
[17] |
R. Gerstberger and P. Rentrop,
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M. C. Grant and S. P. Boyd, Graph implementations for nonsmooth convex programs, Recent Advances in Learning and Control, V. Blondel, S. Boyd, and H. Kimura, eds., Lecture Notes in Control and Information Sciences, Springer-Verlag Limited, 371 (2008), 95–110.
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[19] |
M. Grant and S. Boyd, CVX: Matlab Software for Disciplined Convex Programming, version 2.1., http://cvxr.com/cvx, Mar. 2014. Google Scholar |
[20] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[21] |
T. Kirchdoerfer and M. Ortiz,
Data-driven computational mechanics, Comput. Methods Appl. Mech. Engrg., 304 (2016), 81-101.
doi: 10.1016/j.cma.2016.02.001. |
[22] |
F. Lu, M. Maggioni and S. Tang, Learning interaction kernels in heterogeneous systems of agents from multiple trajectories, arXiv: 1910.04832. Google Scholar |
[23] |
F. Lu, M. Zhong, S. Tang and M. Maggioni,
Nonparametric inference of interaction laws in systems of agents from trajectory data, Proc. Natl. Acad. Sci. USA, 116 (2019), 14424-14433.
doi: 10.1073/pnas.1822012116. |
[24] |
E. Novak and H. Woźniakowski,
Approximation of infinitely differentiable multivariate functions is intractable, J. Complexity, 25 (2009), 398-404.
doi: 10.1016/j.jco.2008.11.002. |
[25] |
H. Schaeffer, G. Tran and R. Ward,
Extracting sparse high-dimensional dynamics from limited data, SIAM J. Appl. Math., 78 (2018), 3279-3295.
doi: 10.1137/18M116798X. |
[26] |
H. Schaeffer, G. Tran and R. Ward, Learning dynamical systems and bifurcation via group sparsity, arXiv: 1709.01558. Google Scholar |
[27] |
G. Scilla and F. Solombrino,
Delayed loss of stability in singularly perturbed finite-dimensional gradient flows, Asymptot. Anal., 110 (2018), 1-19.
doi: 10.3233/ASY-181475. |
[28] |
G. Scilla and F. Solombrino,
Multiscale analysis of singularly perturbed finite dimensional gradient flows: The minimizing movement approach, Nonlinearity, 31 (2018), 5036-5074.
doi: 10.1088/1361-6544/aad6ac. |
[29] |
G. Tran and R. Ward,
Exact recovery of chaotic systems from highly corrupted data, Multiscale Model. Simul., 15 (2017), 1108-1129.
doi: 10.1137/16M1086637. |
[30] |
C. Zanini,
Singular perturbations of finite dimensional gradient flows, Discrete Contin. Dyn. Syst., 18 (2007), 657-675.
doi: 10.3934/dcds.2007.18.657. |
[31] |
M. Zhong, J. Miller and M. Maggioni, Data-driven discovery of emergent behaviors in collective dynamics, Phys. D, 411 (2020), 132542, 25 pp.
doi: 10.1016/j.physd.2020.132542. |
show all references
References:
[1] |
V. Agostiniani and R. Rossi,
Singular vanishing-viscosity limits of gradient flows: The finite-dimensional case, J. Differential Equations, 263 (2017), 7815-7855.
doi: 10.1016/j.jde.2017.08.027. |
[2] |
V. Agostiniani, R. Rossi and G. Savaré,
On the transversality conditions and their genericity, Rend. Circ. Mat. Palermo, 64 (2015), 101-116.
doi: 10.1007/s12215-014-0184-4. |
[3] |
V. Agostiniani, R. Rossi, and G. Savaré, Singular vanishing-viscosity limits of gradient flows in Hilbert spaces, personal communication: in preparation, (2018). Google Scholar |
[4] |
V. Albani, U. M. Ascher, X. Yang and J. P. Zubelli,
Data driven recovery of local volatility surfaces, Inverse Probl. Imaging, 11 (2017), 799-823.
doi: 10.3934/ipi.2017038. |
[5] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. |
[6] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, second ed., 2008. |
[7] |
M. Bongini, M. Fornasier, M. Hansen and M. Maggioni,
Inferring interaction rules from observations of evolutive systems Ⅰ: The variational approach, Math. Models Methods Appl. Sci., 27 (2017), 909-951.
doi: 10.1142/S0218202517500208. |
[8] |
A. Braides, Γ-Convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications, 22. Oxford University Press, Oxford, 2002.
doi: 10.1093/acprof:oso/9780198507840.001.0001. |
[9] |
T. Q. Chen, Y. Rubanova, J. Bettencourt and D. K. Duvenaud, Neural ordinary differential equations, Advances in Neural Information Processing Systems, S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi and R. Garnett, eds., Curran Associates, Inc., 31 (2018), 6571–6583. Google Scholar |
[10] |
S. Conti, S. Müller and M. Ortiz,
Data-driven problems in elasticity, Arch. Ration. Mech. Anal., 229 (2018), 79-123.
doi: 10.1007/s00205-017-1214-0. |
[11] |
S. Crépey,
Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization, SIAM J. Math. Anal., 34 (2003), 1183-1206.
doi: 10.1137/S0036141001400202. |
[12] |
T. S. Cubitt, J. Eisert and M. M. Wolf, Extracting dynamical equations from experimental data is NP hard, Phys. Rev. Lett., 108 (2012), 120503.
doi: 10.1103/PhysRevLett.108.120503. |
[13] |
G. Dal Maso, An Introduction to $\Gamma$-Convergence, vol. 8 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc., Boston, MA, 1993.
doi: 10.1007/978-1-4612-0327-8. |
[14] |
W. E,
A proposal on machine learning via dynamical systems, Commun. Math. Stat., 5 (2017), 1-11.
doi: 10.1007/s40304-017-0103-z. |
[15] |
W. E, J. Han and Q. Li, A mean-field optimal control formulation of deep learning, Res. Math. Sci., 6 (2019), No. 10, 41 pp.
doi: 10.1007/s40687-018-0172-y. |
[16] |
H. Egger and H. W. Engl,
Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates, Inverse Problems, 21 (2005), 1027-1045.
doi: 10.1088/0266-5611/21/3/014. |
[17] |
R. Gerstberger and P. Rentrop,
Feedforward neural nets as discretization schemes for ODES and DAES, 7th ICCAM 96 Congress., J. Comput. Appl. Math., 82 (1997), 117-128.
doi: 10.1016/S0377-0427(97)00085-X. |
[18] |
M. C. Grant and S. P. Boyd, Graph implementations for nonsmooth convex programs, Recent Advances in Learning and Control, V. Blondel, S. Boyd, and H. Kimura, eds., Lecture Notes in Control and Information Sciences, Springer-Verlag Limited, 371 (2008), 95–110.
doi: 10.1007/978-1-84800-155-8_7. |
[19] |
M. Grant and S. Boyd, CVX: Matlab Software for Disciplined Convex Programming, version 2.1., http://cvxr.com/cvx, Mar. 2014. Google Scholar |
[20] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[21] |
T. Kirchdoerfer and M. Ortiz,
Data-driven computational mechanics, Comput. Methods Appl. Mech. Engrg., 304 (2016), 81-101.
doi: 10.1016/j.cma.2016.02.001. |
[22] |
F. Lu, M. Maggioni and S. Tang, Learning interaction kernels in heterogeneous systems of agents from multiple trajectories, arXiv: 1910.04832. Google Scholar |
[23] |
F. Lu, M. Zhong, S. Tang and M. Maggioni,
Nonparametric inference of interaction laws in systems of agents from trajectory data, Proc. Natl. Acad. Sci. USA, 116 (2019), 14424-14433.
doi: 10.1073/pnas.1822012116. |
[24] |
E. Novak and H. Woźniakowski,
Approximation of infinitely differentiable multivariate functions is intractable, J. Complexity, 25 (2009), 398-404.
doi: 10.1016/j.jco.2008.11.002. |
[25] |
H. Schaeffer, G. Tran and R. Ward,
Extracting sparse high-dimensional dynamics from limited data, SIAM J. Appl. Math., 78 (2018), 3279-3295.
doi: 10.1137/18M116798X. |
[26] |
H. Schaeffer, G. Tran and R. Ward, Learning dynamical systems and bifurcation via group sparsity, arXiv: 1709.01558. Google Scholar |
[27] |
G. Scilla and F. Solombrino,
Delayed loss of stability in singularly perturbed finite-dimensional gradient flows, Asymptot. Anal., 110 (2018), 1-19.
doi: 10.3233/ASY-181475. |
[28] |
G. Scilla and F. Solombrino,
Multiscale analysis of singularly perturbed finite dimensional gradient flows: The minimizing movement approach, Nonlinearity, 31 (2018), 5036-5074.
doi: 10.1088/1361-6544/aad6ac. |
[29] |
G. Tran and R. Ward,
Exact recovery of chaotic systems from highly corrupted data, Multiscale Model. Simul., 15 (2017), 1108-1129.
doi: 10.1137/16M1086637. |
[30] |
C. Zanini,
Singular perturbations of finite dimensional gradient flows, Discrete Contin. Dyn. Syst., 18 (2007), 657-675.
doi: 10.3934/dcds.2007.18.657. |
[31] |
M. Zhong, J. Miller and M. Maggioni, Data-driven discovery of emergent behaviors in collective dynamics, Phys. D, 411 (2020), 132542, 25 pp.
doi: 10.1016/j.physd.2020.132542. |



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