Neural Jump Stochastic Differential Equations

By Junteng Jia, Austin R. Benson. 2019

281 citations recorded by the source corpus.

Many time series are effectively generated by a combination of deterministic continuous flows along with discrete jumps sparked by stochastic events. However, we usually do not have the equation of motion describing the flows, or how they are affected by jumps. To this end, we introduce Neural Jump Stochastic Differential Equations that provide a data-driven approach to learn continuous and discrete dynamic behavior, i.e., hybrid systems that both flow and jump. Our approach extends the framework of Neural Ordinary Differential Equations with a stochastic process term that models discrete events. We then model temporal point processes with a piecewise-continuous latent trajectory, where the discontinuities are caused by stochastic events whose conditional intensity depends on the latent state. We demonstrate the predictive capabilities of our model on a range of synthetic and real-world marked point process datasets, including classical point processes (such as Hawkes processes), awards on Stack Overflow, medical records, and earthquake monitoring.

Subfields: Finance, Statistical and Nonlinear Physics, Statistics and Probability, Model Architectures, Probabilistic & Bayesian Methods

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Citations in the corpus

Stochastic Normalizing Flows (2020, background): “Neural stochastic differential equations learn optimal parameters of designed stochastic processes from observations along the path [43, 19, 27, 26], but are not designed for marginal density estimation or asymptotically unbiased sampling.”
Scalable Gradients for Stochastic Differential Equations (2020, background): “Recently, it was combined with modern reverse-mode automatic diﬀerentiation packages, enabling ODEs with millions of parameters to be ﬁt to data [11] and allowing more ﬂexible density estimation and time-series models [22, 31, 69].”
Disentangling Physical Dynamics From Unknown Factors for Unsupervised Video Prediction (2020, unknown): “with stochastic differential equations [23].”
Neural Controlled Differential Equations for Irregular Time Series (2020, background): “[35] consider something similar by interrupting a Neural ODE with stochastic events.”
Liquid Time-constant Networks (2020, background): “2019) and to characterize them better (Dupont, Doucet, and Teh 2019; Durkan et al. 2019; Jia and Benson 2019; Hanshu et al. 2020; Holl, Koltun, and Thuerey 2020; Quaglino et al. 2020).”
Stochastic Normalizing Flows (2020, background): “Stochastic extensions of neural ODEs have been considered in (Tzen & Raginsky, 2019; Liu et al., 2019; Jia & Benson, 2019; Peluchetti & Favaro, 2019) as limits of deep latent Gaussian models, where they have been suggested to show increased robustness to noisy / adversarial data.”
Continuous-in-Depth Neural Networks (2020, background): “Other theoretical research fronts include interfacing Neural ODEs with normalizing flows [22, 79, 40, 56], graph NNs [61, 67], stochastic differential equations [52, 37, 24, 47] and RNNs [10, 65, 27, 18, 49].”