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April  2022, 15(4): 747-771. doi: 10.3934/dcdss.2021103

ISALT: Inference-based schemes adaptive to large time-stepping for locally Lipschitz ergodic systems

Xingjie Helen Li 1, , Fei Lu 2,, and  Felix X.-F. Ye 3,

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

* Corresponding author: Fei Lu

Received  February 2021 Revised  June 2021 Published  April 2022 Early access  September 2021

Fund Project: XL is supported by NSF DMS CAREER-1847770. FL is supported NSF DMS 1913243 and NSF DMS 1821211. FY is supported by AMS-Simons travel grants

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.

Keywords: Stochastic differential equations, inference-based scheme, model reduction in time, locally Lipschitz ergodic systems, data-driven modeling.
Mathematics Subject Classification: Primary: 65C30, 60H35; Secondary: 37M25, 62M20.
Citation: Xingjie Helen Li, Fei Lu, Felix X.-F. Ye. ISALT: Inference-based schemes adaptive to large time-stepping for locally Lipschitz ergodic systems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 747-771. doi: 10.3934/dcdss.2021103
References:
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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. 
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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.

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C. C. Heyde, On the central limit theorem for stationary processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 30 (1974), 315-320.  doi: 10.1007/BF00532619.

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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.

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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.

Figure 1.  Schematic plot of inferring explicit scheme with a large time-step
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Figure 2.  Large-time statistics for 1D double-well potential. (a) TVD between the empirical invariant densities (PDF) of the inferred schemes and the reference PDF from data. (b) and (c): PDFs and ACFs comparison between the IS-RK4 with $ c_0 $ excluded and the reference data
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Figure 3.  1D double-well potential: Convergence of estimators in IS-RK4 with $ c_0 $ excluded. (a) The relative error of the estimator $ \widehat{c_1^{{\delta}, N,M}} $ with $ {\delta} = 80\times \Delta t $ converges at an order about $ (MN)^{-1/2} $, matching Theorem 3.5. (b) Left column: The coefficients depend on the time-step $ {\delta} = {\mathrm{Gap}}\times \Delta t $, with $ c_1 $ being almost 1 and $ c_2 $ being close to linear in $ {\delta} $ until $ {\delta}>0.08 $. The error bars, which are too narrow to be seen, are the standard deviations of the single-trajectory estimators from the $ M $-trajectory estimator. Right column: The residual decays at an order $ O({\delta}^{1/2}) $, matching Theorem 3.6
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Figure 4.  Large-time statistics for the 2D gradient system. (a) TVD between the $ x_1 $ marginal invariant densities (PDF) of the inferred schemes and the reference PDF from data. (b) and (c): PDFs and ACFs comparison between IS-SSBE with $ c_0 $ excluded and the reference data
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Figure 5.  2D gradient system: Convergence of estimators in IS-SSBE with $ c_0 $ excluded. (a) The relative error of the estimator $ \widehat{c_1^{{\delta}, N,M}} $ with $ {\delta} = 120 \Delta t $ converges at an order about $ (MN)^{-1/2} $, matching Theorem 3.5. (b) Left column: The estimators of $ c_1, c_2 $ are almost linear in $ {\delta} $. Right column: The residual changes little as $ {\delta} $ decreases, due to that IS-SSBE is not a parametrization of an explicit scheme (thus, Theorem 3.6 does not apply)
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Figure 6.  2D gradient system: Convergence of estimators in IS-RK4 with $ c_0 $ excluded. (a) The relative error of the estimator $ \widehat{c_1^{{\delta}, N,M}} $ with $ {\delta} = 120 \Delta t $ converges at an order about $ (MN)^{-1/2} $, matching Theorem 3.5. (b) Left column: The estimators of $ c_1, c_2 $ are constant for all $ {\delta} $. Right column: The residual decays at an order $ O({\delta}^{1/2}) $, matching Theorem 3.6
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Figure 7.  Large-time statistics of $ x_1 $ for the stochastic Lorenz system. (a) TVD between the $ x_1 $ marginal invariant densities (PDF) of the inferred schemes and the reference PDF from data. (b) and (c): PDFs and ACFs comparison between IS-RK4 with $ c_0 $ included and the reference data
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Figure 8.  ACF and PDF of $ x_3 $ in the stochastic Lorenz system. Similar to the other examples, IS-RK4 (with $ c_0 $ included) reproduces the PDF and the ACF the best when the time-step is medium large, while plain RK4 and IS-EM blow up even when $ {\mathrm{Gap}} = 20 $
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Figure 9.  The 3D stochastic Lorenz system: Convergence of estimators in IS-RK4 with $ c_0 $ included. (a) The relative error of the estimator $ \widehat{c_1^{{\delta}, N,M}} $ with $ {\delta} = 240 \Delta t = 0.12 $ converges at order about $ (MN)^{-1/2} $, matching Theorem 3.5. (b) Left column: The estimators of $ c_0,c_1, c_2 $ are varies little until $ {\delta}>0.12 $. The vertical dash line is the optimal time gap. Right column: The residuals decay at orders slightly higher than $ O({\delta}^{1/2}) $
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Table 1.  Notations
Notation Description
$ {{\bf X}}_t $ and $ {{\bf B}}_t $ true state process and original stochastic force
$ f({{\bf X}}_t) $, $ \sigma\in \mathbb{R}^{d\times m} $ local-Lipschitz drift and diffusion matrix
$ dt $ time-step generating data
$ {\delta}= {\mathrm{Gap}} \times dt $ time-step for inferred scheme, $ {\mathrm{Gap}}\in \{ 1, 2, 4, 10, 20, 40,\ldots\} $
$ t_i = i{\delta} $ discrete time instants of data
$ \{{{\bf X}}_{t_0:t_N}^{(m)}, {{\bf B}}_{t_0:t_N}^{(m)}\}_{m=1}^M $ Data: $ M $ independent paths of $ {{\bf X}} $ and $ {{\bf B}} $ at discrete-times
$ \mathcal{F}\left({{\bf X}}_{t_i},\, {{\bf B}}_{[t_{i}, \, t_{i+1})}\right) $ true flow map representing $ ({{\bf X}}_{t_{i+1}}-{{\bf X}}_{t_i})/{\delta} $
$ {F}^{\delta}({{\bf X}}_{t_n},\Delta {{\bf B}}_{t_n}) $ approximate flow map using only $ {{\bf X}}_{t_n} $, $ \Delta {{\bf B}}_{t_n} = {{\bf B}}_{t_{n+1}}-{{\bf B}}_{t_{n}} $
$ \widetilde F^{\delta}\left(c^{\delta}, {{\bf X}}_{t_n},\Delta {{\bf B}}_{t_n} \right) $ parametric approximate flow map
$ c^{\delta}=(c_0^{\delta},\dots,c_p^{\delta}) $ parameters to be estimated for the inferred scheme
$ \eta_n $ and $ \sigma_{\eta}^{\delta} $ iid $ N(0, I_d) $ and covariance, representing regression residual
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
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Notation Description
$ {{\bf X}}_t $ and $ {{\bf B}}_t $ true state process and original stochastic force
$ f({{\bf X}}_t) $, $ \sigma\in \mathbb{R}^{d\times m} $ local-Lipschitz drift and diffusion matrix
$ dt $ time-step generating data
$ {\delta}= {\mathrm{Gap}} \times dt $ time-step for inferred scheme, $ {\mathrm{Gap}}\in \{ 1, 2, 4, 10, 20, 40,\ldots\} $
$ t_i = i{\delta} $ discrete time instants of data
$ \{{{\bf X}}_{t_0:t_N}^{(m)}, {{\bf B}}_{t_0:t_N}^{(m)}\}_{m=1}^M $ Data: $ M $ independent paths of $ {{\bf X}} $ and $ {{\bf B}} $ at discrete-times
$ \mathcal{F}\left({{\bf X}}_{t_i},\, {{\bf B}}_{[t_{i}, \, t_{i+1})}\right) $ true flow map representing $ ({{\bf X}}_{t_{i+1}}-{{\bf X}}_{t_i})/{\delta} $
$ {F}^{\delta}({{\bf X}}_{t_n},\Delta {{\bf B}}_{t_n}) $ approximate flow map using only $ {{\bf X}}_{t_n} $, $ \Delta {{\bf B}}_{t_n} = {{\bf B}}_{t_{n+1}}-{{\bf B}}_{t_{n}} $
$ \widetilde F^{\delta}\left(c^{\delta}, {{\bf X}}_{t_n},\Delta {{\bf B}}_{t_n} \right) $ parametric approximate flow map
$ c^{\delta}=(c_0^{\delta},\dots,c_p^{\delta}) $ parameters to be estimated for the inferred scheme
$ \eta_n $ and $ \sigma_{\eta}^{\delta} $ iid $ N(0, I_d) $ and covariance, representing regression residual
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
Algorithm 1.  Inference-based schemes adaptive to large time-stepping (ISALT): detailed algorithm
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 $ dt $; down sample to get time series with $ {\delta}= \mathrm{Gap}\times dt $. Denote the data, consisting of $ M $ independent trajectories on $ [0,N{\delta}] $, by $ \{{{\bf X}}_{t_0:t_N}^{(m)}, {{\bf B}}_{t_0:t_N}^{(m)}\}_{m=1}^M $ with $ t_i= i{\delta} $.
2: Pick a parametric form approximating the flow map (2.1) as in (2.5)–(2.6).
3: Estimate parameters $ c_{0:p}^{\delta} $ and $ \sigma_\eta $ as in (2.7).
4: Model selection: run the inferred scheme for cross-validation, and test the consistency of the estimators.
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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 $ dt $; down sample to get time series with $ {\delta}= \mathrm{Gap}\times dt $. Denote the data, consisting of $ M $ independent trajectories on $ [0,N{\delta}] $, by $ \{{{\bf X}}_{t_0:t_N}^{(m)}, {{\bf B}}_{t_0:t_N}^{(m)}\}_{m=1}^M $ with $ t_i= i{\delta} $.
2: Pick a parametric form approximating the flow map (2.1) as in (2.5)–(2.6).
3: Estimate parameters $ c_{0:p}^{\delta} $ and $ \sigma_\eta $ as in (2.7).
4: Model selection: run the inferred scheme for cross-validation, and test the consistency of the estimators.
Table 2.  Time gap of blow-up for each scheme: plain verse inferred
1D double-well 2D gradient system 3D Lorenz system
Plain RK4 $ {\mathrm{Gap}}=20 $ $ {\mathrm{Gap}}=20 $ $ {\mathrm{Gap}}=10 $
IS-RK4 $ {\mathrm{Gap}}>200 $ $ {\mathrm{Gap}}>200 $ $ {\mathrm{Gap}}>400 $
Plain SSBE $ {\mathrm{Gap}}=40 $ $ {\mathrm{Gap}}=40 $ $ {\mathrm{Gap}}=20 $
IS-SSBE $ {\mathrm{Gap}}>200 $ $ {\mathrm{Gap}}>200 $ $ {\mathrm{Gap}}>400 $
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1D double-well 2D gradient system 3D Lorenz system
Plain RK4 $ {\mathrm{Gap}}=20 $ $ {\mathrm{Gap}}=20 $ $ {\mathrm{Gap}}=10 $
IS-RK4 $ {\mathrm{Gap}}>200 $ $ {\mathrm{Gap}}>200 $ $ {\mathrm{Gap}}>400 $
Plain SSBE $ {\mathrm{Gap}}=40 $ $ {\mathrm{Gap}}=40 $ $ {\mathrm{Gap}}=20 $
IS-SSBE $ {\mathrm{Gap}}>200 $ $ {\mathrm{Gap}}>200 $ $ {\mathrm{Gap}}>400 $
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