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A drift homotopy implicit particle filter method for nonlinear filtering problems
ISALT: Inference-based schemes adaptive to large time-stepping for locally Lipschitz ergodic systems
1. | Department of Mathematics and Statistics, University of North Carolina at Charlotte, 9201 Univ City Blvd., Charlotte, NC 28023, USA |
2. | Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA |
3. | Department of Mathematics and Statistics, State University of New York at Albany, Earth Science 110, 1400 Washington Avenue, Albany, NY 12222, USA |
Efficient simulation of SDEs is essential in many applications, particularly for ergodic systems that demand efficient simulation of both short-time dynamics and large-time statistics. However, locally Lipschitz SDEs often require special treatments such as implicit schemes with small time-steps to accurately simulate the ergodic measures. We introduce a framework to construct inference-based schemes adaptive to large time-steps (ISALT) from data, achieving a reduction in time by several orders of magnitudes. The key is the statistical learning of an approximation to the infinite-dimensional discrete-time flow map. We explore the use of numerical schemes (such as the Euler-Maruyama, the hybrid RK4, and an implicit scheme) to derive informed basis functions, leading to a parameter inference problem. We introduce a scalable algorithm to estimate the parameters by least squares, and we prove the convergence of the estimators as data size increases.
We test the ISALT on three non-globally Lipschitz SDEs: the 1D double-well potential, a 2D multiscale gradient system, and the 3D stochastic Lorenz equation with a degenerate noise. Numerical results show that ISALT can tolerate time-step magnitudes larger than plain numerical schemes. It reaches optimal accuracy in reproducing the invariant measure when the time-step is medium-large.
References:
[1] |
Y. Bar-Sinai, S. Hoyer, J. Hickey and M. P. Brenner,
Learning data-driven discretizations for partial differential equations, Proc. Natl. Acad. Sci. USA, 116 (2019), 15344-15349.
doi: 10.1073/pnas.1814058116. |
[2] |
A. J. Chorin and F. Lu,
Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics, Proceedings of the National Academy of Sciences, USA, 112 (2015), 9804-9809.
|
[3] |
A. J. Chorin, F. Lu, R. N. Miller, M. Morzfeld and X. Tu,
Sampling, feasibility, and priors in data assimilation, Discrete Contin. Dyn. Syst., 36 (2016), 4227-4246.
doi: 10.3934/dcds.2016.36.4227. |
[4] |
W. E, B. Engquist, X. Li, W. Ren and E. Vanden-Eijnden,
The heterogeneous multiscale method: A review, Commun. Comput. Phys., 2 (2007), 367-450.
|
[5] |
P. Hall and C. C. Heyde, Martingale Limit Theory and its Application, Academic press, 1980.
![]() ![]() |
[6] |
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. |
[7] |
J. A. Hansen and C. Penland,
Efficient approximate technique for integrating stochastic differential equations, Monthly Weather Review, 134 (2006), 3006-3014.
doi: 10.1175/MWR3192.1. |
[8] |
C. C. Heyde,
On the central limit theorem for stationary processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 30 (1974), 315-320.
doi: 10.1007/BF00532619. |
[9] |
Y. Hu, Strong and weak order of time discretization schemes of stochastic differential equatios, In Séminaire de Probabilités XXX, Springer, (1996), 218–227.
doi: 10.1007/BFb0094650. |
[10] |
T. Hudson and X. H. Li,
Coarse-graining of overdamped Langevin dynamics via the Mori–Zwanzig formalism, Multiscale Model. Simul., 18 (2020), 1113-1135.
doi: 10.1137/18M1222533. |
[11] |
M. Hutzenthaler and A. Jentzen, Numerical Approximations of Stochastic Differential Equations with Non-globally Lipschitz Continuous Coefficients, American Mathematical Society, 2015.
doi: 10.1090/memo/1112. |
[12] |
M. Hutzenthaler, A. Jentzen and P. E. Kloeden,
Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab., 22 (2012), 1611-1641.
doi: 10.1214/11-AAP803. |
[13] |
A. Jentzen and P. Kloeden,
Taylor expansions of solutions of stochastic partial differential equations with additive noise, Ann. Probab., 38 (2010), 532-569.
doi: 10.1214/09-AOP500. |
[14] |
S. W. Jiang and J. Harlim, Modeling of missing dynamical systems: Deriving parametric models using a nonparametric framework, Res. Math. Sci., 7 (2020), Paper No. 16, 25 pp.
doi: 10.1007/s40687-020-00217-4. |
[15] |
R. Khasminskii, Stochastic Stability of Differential Equations, volume 66., Springer-Verlag Berlin Heidelberg, 2nd edition, 2012.
doi: 10.1007/978-3-642-23280-0. |
[16] |
B. Khouider, A. J. Majda and M. A. Katsoulakis,
Coarse-grained stochastic models for tropical convection and climate, Proc. Natl. Acad. Sci. USA, 100 (2003), 11941-11946.
|
[17] |
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 3rd edition, 1992.
doi: 10.1007/978-3-662-12616-5. |
[18] |
K. Law, A. Stuart and K. Zygalakis, Data Assimilation: A Mathematical Introduction, Springer, 2015.
doi: 10.1007/978-3-319-20325-6. |
[19] |
F. Legoll and T. Lelièvre,
Effective dynamics using conditional expectations, Nonlinearity, 23 (2010), 2131-2163.
doi: 10.1088/0951-7715/23/9/006. |
[20] |
F. Legoll, T. Leliévre and U. Sharma,
Effective dynamics for non-reversible stochastic differential equations: A quantitative study, Nonlinearity, 32 (2019), 4779-4816.
doi: 10.1088/1361-6544/ab34bf. |
[21] |
H. Lei, N. A. Baker and X. Li,
Data-driven parameterization of the generalized Langevin equation, Proc. Natl. Acad. Sci. USA, 113 (2016), 14183-14188.
|
[22] |
B. Leimkuhler and C. Matthews, Molecular Dynamics, Springer, 2015. |
[23] |
Y. Li and J. Duan, A data-driven approach for discovering stochastic dynamical systems with non-Gaussian Lévy noise, Phys. D, 417 (2021), 132830, 12 pp.
doi: 10.1016/j.physd.2020.132830. |
[24] |
K. K. Lin and F. Lu, Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism, J. Comput. Phys., 424 (2021), 109864, 33 pp.
doi: 10.1016/j.jcp.2020.109864. |
[25] |
S. Liu, L. Grzelak and C. W. Oosterlee, The seven-league scheme: Deep learning for large time step monte carlo simulations of stochastic differential equations, arXiv: 2009.03202, (2020). |
[26] |
F. Lu, Data-driven model reduction for stochastic Burgers equations, Entropy, 22 (2020), Paper No. 1360, 22 pp.
doi: 10.3390/e22121360. |
[27] |
F. Lu, K. K. Lin and A. J. Chorin,
Comparison of continuous and discrete-time data-based modeling for hypoelliptic systems, Commun. Appl. Math. Comput. Sci., 11 (2016), 187-216.
doi: 10.2140/camcos.2016.11.187. |
[28] |
F. Lu, K. K. Lin and A. J. Chorin,
Data-based stochastic model reduction for the Kuramoto–Sivashinsky equation, Phys. D, 340 (2017), 46-57.
doi: 10.1016/j.physd.2016.09.007. |
[29] |
F. Lu, M. Maggioni and S. Tang, Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories, J. Mach. Learn. Res., 22 (2021), Paper No. 32, 67 pp. |
[30] |
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.
|
[31] |
Y. Maday and G. Turinici,
A parareal in time procedure for the control of partial differential equations, C. R. Math. Acad. Sci. Paris, 335 (2002), 387-392.
doi: 10.1016/S1631-073X(02)02467-6. |
[32] |
A. J. Majda and J. Harlim,
Physics constrained nonlinear regression models for time series, Nonlinearity, 26 (2013), 201-217.
doi: 10.1088/0951-7715/26/1/201. |
[33] |
X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007. |
[34] |
J. C. Mattingly, A. M. Stuart and and D. J. Higham,
Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise, Stochastic Process. Appl., 101 (2002), 185-232.
doi: 10.1016/S0304-4149(02)00150-3. |
[35] |
G. A. Pavliotis and A. M. Stuart,
Parameter estimation for multiscale diffusions, J. Statist. Phys., 127 (2007), 741-781.
doi: 10.1007/s10955-007-9300-6. |
[36] |
G. O. Roberts and R. L. Tweedie,
Exponential convergence of Langevin distributions and their discrete approximations, Bernoulli, 2 (1996), 341-363.
doi: 10.2307/3318418. |
[37] |
W. Rümelin,
Numerical treatment of stochastic differential equations, SIAM J. Numer. Anal., 19 (1982), 604-613.
doi: 10.1137/0719041. |
[38] |
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. |
[39] |
L. Yang, D. Zhang and G. E. Karniadakis,
Physics-informed generative adversarial networks for stochastic differential equations, SIAM J. Sci. Comput., 42 (2020), A292-A317.
doi: 10.1137/18M1225409. |
show all references
References:
[1] |
Y. Bar-Sinai, S. Hoyer, J. Hickey and M. P. Brenner,
Learning data-driven discretizations for partial differential equations, Proc. Natl. Acad. Sci. USA, 116 (2019), 15344-15349.
doi: 10.1073/pnas.1814058116. |
[2] |
A. J. Chorin and F. Lu,
Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics, Proceedings of the National Academy of Sciences, USA, 112 (2015), 9804-9809.
|
[3] |
A. J. Chorin, F. Lu, R. N. Miller, M. Morzfeld and X. Tu,
Sampling, feasibility, and priors in data assimilation, Discrete Contin. Dyn. Syst., 36 (2016), 4227-4246.
doi: 10.3934/dcds.2016.36.4227. |
[4] |
W. E, B. Engquist, X. Li, W. Ren and E. Vanden-Eijnden,
The heterogeneous multiscale method: A review, Commun. Comput. Phys., 2 (2007), 367-450.
|
[5] |
P. Hall and C. C. Heyde, Martingale Limit Theory and its Application, Academic press, 1980.
![]() ![]() |
[6] |
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. |
[7] |
J. A. Hansen and C. Penland,
Efficient approximate technique for integrating stochastic differential equations, Monthly Weather Review, 134 (2006), 3006-3014.
doi: 10.1175/MWR3192.1. |
[8] |
C. C. Heyde,
On the central limit theorem for stationary processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 30 (1974), 315-320.
doi: 10.1007/BF00532619. |
[9] |
Y. Hu, Strong and weak order of time discretization schemes of stochastic differential equatios, In Séminaire de Probabilités XXX, Springer, (1996), 218–227.
doi: 10.1007/BFb0094650. |
[10] |
T. Hudson and X. H. Li,
Coarse-graining of overdamped Langevin dynamics via the Mori–Zwanzig formalism, Multiscale Model. Simul., 18 (2020), 1113-1135.
doi: 10.1137/18M1222533. |
[11] |
M. Hutzenthaler and A. Jentzen, Numerical Approximations of Stochastic Differential Equations with Non-globally Lipschitz Continuous Coefficients, American Mathematical Society, 2015.
doi: 10.1090/memo/1112. |
[12] |
M. Hutzenthaler, A. Jentzen and P. E. Kloeden,
Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab., 22 (2012), 1611-1641.
doi: 10.1214/11-AAP803. |
[13] |
A. Jentzen and P. Kloeden,
Taylor expansions of solutions of stochastic partial differential equations with additive noise, Ann. Probab., 38 (2010), 532-569.
doi: 10.1214/09-AOP500. |
[14] |
S. W. Jiang and J. Harlim, Modeling of missing dynamical systems: Deriving parametric models using a nonparametric framework, Res. Math. Sci., 7 (2020), Paper No. 16, 25 pp.
doi: 10.1007/s40687-020-00217-4. |
[15] |
R. Khasminskii, Stochastic Stability of Differential Equations, volume 66., Springer-Verlag Berlin Heidelberg, 2nd edition, 2012.
doi: 10.1007/978-3-642-23280-0. |
[16] |
B. Khouider, A. J. Majda and M. A. Katsoulakis,
Coarse-grained stochastic models for tropical convection and climate, Proc. Natl. Acad. Sci. USA, 100 (2003), 11941-11946.
|
[17] |
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 3rd edition, 1992.
doi: 10.1007/978-3-662-12616-5. |
[18] |
K. Law, A. Stuart and K. Zygalakis, Data Assimilation: A Mathematical Introduction, Springer, 2015.
doi: 10.1007/978-3-319-20325-6. |
[19] |
F. Legoll and T. Lelièvre,
Effective dynamics using conditional expectations, Nonlinearity, 23 (2010), 2131-2163.
doi: 10.1088/0951-7715/23/9/006. |
[20] |
F. Legoll, T. Leliévre and U. Sharma,
Effective dynamics for non-reversible stochastic differential equations: A quantitative study, Nonlinearity, 32 (2019), 4779-4816.
doi: 10.1088/1361-6544/ab34bf. |
[21] |
H. Lei, N. A. Baker and X. Li,
Data-driven parameterization of the generalized Langevin equation, Proc. Natl. Acad. Sci. USA, 113 (2016), 14183-14188.
|
[22] |
B. Leimkuhler and C. Matthews, Molecular Dynamics, Springer, 2015. |
[23] |
Y. Li and J. Duan, A data-driven approach for discovering stochastic dynamical systems with non-Gaussian Lévy noise, Phys. D, 417 (2021), 132830, 12 pp.
doi: 10.1016/j.physd.2020.132830. |
[24] |
K. K. Lin and F. Lu, Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism, J. Comput. Phys., 424 (2021), 109864, 33 pp.
doi: 10.1016/j.jcp.2020.109864. |
[25] |
S. Liu, L. Grzelak and C. W. Oosterlee, The seven-league scheme: Deep learning for large time step monte carlo simulations of stochastic differential equations, arXiv: 2009.03202, (2020). |
[26] |
F. Lu, Data-driven model reduction for stochastic Burgers equations, Entropy, 22 (2020), Paper No. 1360, 22 pp.
doi: 10.3390/e22121360. |
[27] |
F. Lu, K. K. Lin and A. J. Chorin,
Comparison of continuous and discrete-time data-based modeling for hypoelliptic systems, Commun. Appl. Math. Comput. Sci., 11 (2016), 187-216.
doi: 10.2140/camcos.2016.11.187. |
[28] |
F. Lu, K. K. Lin and A. J. Chorin,
Data-based stochastic model reduction for the Kuramoto–Sivashinsky equation, Phys. D, 340 (2017), 46-57.
doi: 10.1016/j.physd.2016.09.007. |
[29] |
F. Lu, M. Maggioni and S. Tang, Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories, J. Mach. Learn. Res., 22 (2021), Paper No. 32, 67 pp. |
[30] |
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.
|
[31] |
Y. Maday and G. Turinici,
A parareal in time procedure for the control of partial differential equations, C. R. Math. Acad. Sci. Paris, 335 (2002), 387-392.
doi: 10.1016/S1631-073X(02)02467-6. |
[32] |
A. J. Majda and J. Harlim,
Physics constrained nonlinear regression models for time series, Nonlinearity, 26 (2013), 201-217.
doi: 10.1088/0951-7715/26/1/201. |
[33] |
X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007. |
[34] |
J. C. Mattingly, A. M. Stuart and and D. J. Higham,
Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise, Stochastic Process. Appl., 101 (2002), 185-232.
doi: 10.1016/S0304-4149(02)00150-3. |
[35] |
G. A. Pavliotis and A. M. Stuart,
Parameter estimation for multiscale diffusions, J. Statist. Phys., 127 (2007), 741-781.
doi: 10.1007/s10955-007-9300-6. |
[36] |
G. O. Roberts and R. L. Tweedie,
Exponential convergence of Langevin distributions and their discrete approximations, Bernoulli, 2 (1996), 341-363.
doi: 10.2307/3318418. |
[37] |
W. Rümelin,
Numerical treatment of stochastic differential equations, SIAM J. Numer. Anal., 19 (1982), 604-613.
doi: 10.1137/0719041. |
[38] |
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. |
[39] |
L. Yang, D. Zhang and G. E. Karniadakis,
Physics-informed generative adversarial networks for stochastic differential equations, SIAM J. Sci. Comput., 42 (2020), A292-A317.
doi: 10.1137/18M1225409. |









Notation | Description |
true state process and original stochastic force | |
local-Lipschitz drift and diffusion matrix | |
time-step generating data | |
time-step for inferred scheme, |
|
discrete time instants of data | |
Data: |
|
true flow map representing |
|
approximate flow map using only |
|
parametric approximate flow map | |
parameters to be estimated for the inferred scheme | |
iid |
|
EM and IS-EM | Euler-Maruyama and inferred scheme (IS) parametrizing it |
HRK4 and IS-RK4 | hybrid RK4 and inferred scheme parametrizing RK4 |
SSBE and IS-SSBE | split-step stochastic backward Euler and IS parametrizing it |
Notation | Description |
true state process and original stochastic force | |
local-Lipschitz drift and diffusion matrix | |
time-step generating data | |
time-step for inferred scheme, |
|
discrete time instants of data | |
Data: |
|
true flow map representing |
|
approximate flow map using only |
|
parametric approximate flow map | |
parameters to be estimated for the inferred scheme | |
iid |
|
EM and IS-EM | Euler-Maruyama and inferred scheme (IS) parametrizing it |
HRK4 and IS-RK4 | hybrid RK4 and inferred scheme parametrizing RK4 |
SSBE and IS-SSBE | split-step stochastic backward Euler and IS parametrizing it |
Input: Full model; a high fidelity solver preserving the invariant measure. Output: Estimated parametric scheme 1: Generate data: solve the system with the high fidelity solver, which has a small time-step 2: Pick a parametric form approximating the flow map (2.1) as in (2.5)–(2.6). 3: Estimate parameters 4: Model selection: run the inferred scheme for cross-validation, and test the consistency of the estimators. |
Input: Full model; a high fidelity solver preserving the invariant measure. Output: Estimated parametric scheme 1: Generate data: solve the system with the high fidelity solver, which has a small time-step 2: Pick a parametric form approximating the flow map (2.1) as in (2.5)–(2.6). 3: Estimate parameters 4: Model selection: run the inferred scheme for cross-validation, and test the consistency of the estimators. |
1D double-well | 2D gradient system | 3D Lorenz system | |
Plain RK4 | |||
IS-RK4 | |||
Plain SSBE | |||
IS-SSBE |
1D double-well | 2D gradient system | 3D Lorenz system | |
Plain RK4 | |||
IS-RK4 | |||
Plain SSBE | |||
IS-SSBE |
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