Abstract
We present a numerical approach for modeling unknown dynamical systems using partially observed data, with a focus on biological systems with (relatively) complex dynamical behavior. As an extension of the recently developed deep neural network (DNN) learning methods, our approach is particularly suitable for practical situations when (i) measurement data are available for only a subset of the state variables, and (ii) the system parameters cannot be observed or measured at all. We demonstrate that, with a properly designed DNN structure with memory terms, effective DNN models can be learned from such partially observed data containing hidden parameters. The learned DNN model serves as an accurate predictive tool for system analysis. Through a few representative biological problems, we demonstrate that such DNN models can capture qualitative dynamical behavior changes in the system, such as bifurcations, even when the parameters controlling such behavior changes are completely unknown throughout not only the model learning process but also the system prediction process. The learned DNN model effectively creates a “closed” model involving only the observables when such a closed-form model does not exist mathematically.
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References
Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mané, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viégas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: TensorFlow: large-scale machine learning on heterogeneous systems (2015). https://www.tensorflow.org/. Software available from tensorflow.org
Bongard, J., Lipson, H.: Automated reverse engineering of nonlinear dynamical systems. Proc. Natl. Acad. Sci. USA 104, 9943–9948 (2007)
Chan, S., Elsheikh, A.: A machine learning approach for efficient uncertainty quantification using multiscale methods. J. Comput. Phys. 354, 494–511 (2018)
Chen, Z., Churchill, V., Wu, K., **u, D.: Deep neural network modeling of unknown partial differential equations in nodal space. J. Comput. Phys. 449, 110782 (2022)
Chen, Z., **u, D.: On generalized residual network for deep learning of unknown dynamical systems. J. Comput. Phys. 438, 110362 (2021)
Daniels, B.C., Nemenman, I.: Automated adaptive inference of phenomenological dynamical models. Nature Communications 6, 8133 (2015)
Daniels, B.C., Nemenman, I.: Efficient inference of parsimonious phenomenological models of cellular dynamics using S-systems and alternating regression. PloS One 10, e0119821 (2015)
DeAngelis, D.L., Yurek, S.: Equation-free modeling unravels the behavior of complex ecological systems. Proc. Natl. Acad. Sci. USA 112, 3856–3857 (2015).
E, W.N., Engquist, B., Huang, Z.: Heterogeneous multiscale method: a general methodology for multiscale modeling. Phys. Rev. B 67, 092101 (2003)
Fall, C., Marland, E., Wagner, J., Tyson, J.: Computational Cell Biology. Springer Science+Business Media, Inc., Berlin (2010)
Fu, X., Chang, L.-B., **u, D.: Learning reduced systems via deep neural networks with memory. J. Mach. Learn. Model. Comput. 1, 97–118 (2020)
Fu, X., Mao, W., Chang, L.-B., **u, D.: Modeling unknown dynamical systems with hidden parameters. J. Mach. Learn. Model. Comput. 3, 79-95 (2022)
Giannakis, D., Majda, A.J.: Nonlinear Laplacian spectral analysis for time series with intermittency and low-frequency variability. Proc. Natl. Acad. Sci. USA 109, 2222–2227 (2012)
Gonzalez-Garcia, R., Rico-Martinez, R., Kevrekidis, I.G.: Identification of distributed parameter systems: a neural net based approach. Comput. Chem. Eng. 22, S965–S968 (1998)
Grimm, V., Railsback, S.F.: Individual-Based Modeling and Ecology. Princeton University Press, Princeton (2005)
Han, J., Jentzen, A., E, W.N.: Solving high-dimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci. 115, 8505–8510 (2018)
Hau, D.T., Coiera, E.W.: Learning qualitative models from physiological signals internal accession date only. Technical report (1995)
Hesthaven, J., Ubbiali, S.: Non-intrusive reduced order modeling of nonlinear problems using neural networks. J. Comput. Phys. 363, 55–78 (2018)
Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol. 117, 500–544 (1952).
Karumuri, S., Tripathy, R., Bilionis, I., Panchal, J.: Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks, J. Comp. Phys. 404, 109120 (2019)
Kevrekidis, I.G., Gear, C.W., Hyman, J.M., Kevrekidid, P.G., Runborg, O., Theodoropoulos, C.: Equation-free, coarse-grained multiscale computation: enabling mocroscopic simulators to perform system-level analysis. Commun. Math. Sci. 1, 715–762 (2003)
Khoo, Y., Lu, J., Ying, L.: Solving parametric PDE problems with artificial neural networks. Euro.Jnl. Appl. Math. 32, 21–435 (2018). ar**v:1707.03351
Long, Z., Lu, Y., Dong, B.: PDE-Net 2.0: learning PDEs from data with a numeric-symbolic hybrid deep network. J. Comput. Phys. 399, 108925 (2019). ar**v:1812.04426 [math.ST]
Long, Z., Lu, Y., Ma, X., Dong, B.: PDE-net: learning PDEs from data. Proceedings of the 35th International Conference on Machine Learning. PMLR 80, 3208–3216 (2018)
Maki, L.W., Keizer, J.: Mathematical analysis of a proposed mechanism for oscillatory insulin secretion in perifused hit-15 cells. Bull. Math. Biol. 57, 569–591 (1995)
Mangan, N.M., Brunton, S.L., Proctor, J.L., Kutz, J.N.: Inferring biological networks by sparse identification of nonlinear dynamics. IEEE Trans. Mol. Biol. Multi-Scale Commun. 2, 52–63 (2016)
Mardt, A., Pasquali, L., Wu, H., Noe, F.: VAMPnets for deep learning of molecular kinetics. Nat. Commun. 9, 5 (2018)
Mori, H.: Transport, collective motion, and Brownian motion. Prog. Theor. Phys. 33, 423–455 (1965)
Morris, C., Lecar, H.: Voltage oscillations in the barnacle giant muscle fiber (1981). https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1327511/
Pawar, S., Rahman, S.M., Vaddireddy, H., San, O., Rasheed, A., Vedula, P.: A deep learning enabler for nonintrusive reduced order modeling of fluid flows. Phys. Fluids 31, 085101 (2019)
Perretti, C.T., Munch, S.B., Sugihara, G.: Model-free forecasting outperforms the correct mechanistic model for simulated and experimental data. Proc. Natl. Acad. Sci. USA 110, 5253–5257 (2013). https://doi.org/10.1073/pnas.1216076110
Qin, T., Chen, Z., Jakeman, J., **u, D.: Data-driven learning of non-autonomous systems. SIAM J. Sci. Comput. 43, A1607–A1624 (2021)
Qin, T., Chen, Z., Jakeman, J., **u, D.: Deep learning of parameterized equations with applications to uncertainty quantification. Int. J. Uncertain. Quantif. 11, 63–82 (2021)
Qin, T., Wu, K., **u, D.: Data driven governing equations approximation using deep neural networks. J. Comput. Phys. 395, 620–635 (2019)
Raissi, M.: Deep hidden physics models: deep learning of nonlinear partial differential equations. J. Mach. Learn. Res. 19, 1–24 (2018)
Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics informed deep learning (part I): data-driven solutions of nonlinear partial differential equations. J. Comput. Phys. 378, 686–707 (2019). ar**v:1711.10561
Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics informed deep learning (part II): data-driven discovery of nonlinear partial differential equations (2017). ar**v:1711.10566
Raissi, M., Perdikaris, P., Karniadakis, G.E.: Multistep neural networks for data-driven discovery of nonlinear dynamical systems (2018). ar**v:1801.01236
Ray, D., Hesthaven, J.: An artificial neural network as a troubled-cell indicator. J. Comput. Phys. 367, 166–191 (2018)
Rinzel, J., Ermentrout, G.: Analysis of neural excitability and oscillations. In: Koch, C., Segev, I. (eds.) Methods in Neuronal Modeling, 2nd edn., pp. 251–291. MIT Press, Cambridge (1998)
Rudy, S.H., Kutz, J.N., Brunton, S.L.: Deep learning of dynamics and signal-noise decomposition with time-step** constraints. J. Comput. Phys. 396, 483–506 (2019)
Schmidt, M.D., Lipson, H.: Distilling free-form natural laws from experimental data. Science 324, 81–85 (2009)
Schmidt, M.D., Vallabhajosyula, R.R., Jenkins, J.W., Hood, J.E., Soni, A.S., Wikswo, J.P., Lipson, H.: Automated refinement and inference of analytical models for metabolic networks. Phys. Biol. 8, 055011 (2011)
Sugihara, G., May, R., Ye, H., Hsieh, C., Deyle, E., Fogarty, M., Munch, S.: Detecting causality in complex ecosystems. Science 338, 496–500 (2012)
Sun, Y., Zhang, L., Schaeffer, H.: NeuPDE: neural network based ordinary and partial differential equations for modeling time-dependent data. Proc. Machine Learning Res. 107, 352–372 (2020). ar**v:1908.03190
Tripathy, R., Bilionis, I.: Deep UQ: learning deep neural network surrogate model for high dimensional uncertainty quantification. J. Comput. Phys. 375, 565–588 (2018)
Tsumoto, K., Kitajima, H., Yoshinaga, T., Aihara, K., Kawakami, H.: Bifurcations in Morris-Lecar neuron model. Neurocomputing 69, 293–316 (2006). https://doi.org/10.1016/j.neucom.2005.03.006
Voss, H.U., Kolodner, P., Abel, M., Kurths, J.: Amplitude equations from spatiotemporal binary-fluid convection data. Phys. Rev. Lett. 83, 3422 (1999)
Wang, Q., Ripamonti, N., Hesthaven, J.: Recurrent neural network closure of parametric POD-Galerkin reduced-order models based on the Mori-Zwanzig formalism. J. Comput. Phys. 410, 109402 (2020)
Wang, Y., Shen, Z., Long, Z., Dong, B.: Learning to discretize: solving 1D scalar conservation laws via deep reinforcement learning. Commun. Comput. Phys. 28, 2158–2179 (2020). ar**v:1905.11079
Wood, S.N., Thomas, M.B.: Super-sensitivity to structure in biological models. Proc. R. Soc. B: Biol. Sci. 266, 565–570 (1999). https://doi.org/10.1098/rspb.1999.0673
Wu, K., **u, D.: Data-driven deep learning of partial differential equations in modal space. J. Comput. Phys. 408, 109307 (2020)
Ye, H., Beamish, R.J., Glaser, S.M., Grant, S.C.H., Hsieh, C., Richards, L.J., Schnute, J.T., Sugihara, G.: Equation-free mechanistic ecosystem forecasting using empirical dynamic modeling. Proc. Natl. Acad. Sci. USA 112, E1569–E1576 (2015)
Yodzis, P.: The indeterminacy of ecological interactions as perceived through perturbation. Technical Report (1988)
Zhu, Y., Zabaras, N.: Bayesian deep convolutional encoder-decoder networks for surrogate modeling and uncertainty quantification. J. Comput. Phys. 366, 415–447 (2018)
Zwanzig, R.: Nonlinear generalized Langevin equations. J. Stat. Phys. 9, 215–220 (1973)
Funding
This work was partially supported by the NSF (No. DMS-1813071) (Chou) and the AFSOR (No. FA9550-22-1-0011) (**u).
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Su, WH., Chou, CS. & **u, D. Data-Driven Modeling of Partially Observed Biological Systems. Commun. Appl. Math. Comput. 6, 739–754 (2024). https://doi.org/10.1007/s42967-023-00317-2
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DOI: https://doi.org/10.1007/s42967-023-00317-2