Neural Jump Stochastic Differential Equations (1905.10403v3)

Abstract: 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.

• The paper introduces Neural JSDEs to combine continuous dynamics with discrete stochastic events, enhancing the modeling capabilities of neural ODEs.
• It employs a latent state updated via temporal point processes to enable accurate and efficient gradient computation using the adjoint method.
• Experimental results on synthetic and real-world datasets, including Hawkes processes and earthquake data, demonstrate superior performance over traditional RNN approaches.

An Examination of Neural Jump Stochastic Differential Equations

In contemporary computational modeling, the representation of real-world systems as a combination of continuous and discrete dynamics is increasingly important. The paper "Neural Jump Stochastic Differential Equations" presents an innovative methodology that augments Neural Ordinary Differential Equations (Neural ODEs) by incorporating stochastic elements that account for discrete events, resulting in the Neural Jump Stochastic Differential Equations (Neural JSDEs). This framework provides a robust approach to understanding hybrid systems characterized by continuous flows and discrete jumps.

Core Contributions

The essence of this paper lies in addressing the complexity of integrating discrete stochastic events into a continuous dynamical framework. Neural JSDEs achieve this by employing a latent state vector, $\mathbf{z}(t)$, which evolves continuously. However, its trajectory is subject to abrupt changes instigated by stochastic events. These events are modeled using temporal point processes where the event's conditional intensity is a function of the latent state, and discontinuities in the state are explicitly managed.

The significant advancement here is the adaptation of Neural ODEs, which are inherently limited to continuous transformations, to handle discrete events without losing computational efficiency. This adaptation is critical as it preserves the key advantage of ODE-based models — the ability to use the adjoint method for backpropagation through time with constant memory usage. The paper meticulously explains how the latent state and adjoint state can be updated at these discontinuities, ensuring precise gradient calculations.

Experimental Validation

The authors demonstrated the capabilities of Neural JSDEs through a series of experiments involving both synthetic datasets and real-world scenarios. Notably, they showed that their model could effectively learn the intensity functions of classical point processes, such as Hawkes processes and self-correcting processes, with considerable accuracy. The model's superior performance over traditional methods like RNNs underscores its efficacy in modeling event-driven systems.

Further experiments on datasets such as Stack Overflow badges and medical records showcased the model’s prowess in predicting the types of events, achieving performance comparable to, if not surpassing, existing RNN and LSTM-based models. The flexibility of the Neural JSDEs framework is also illustrated in its capacity to handle real-valued features within events, as demonstrated by the modeling of earthquake data.

Practical and Theoretical Implications

Practically, Neural JSDEs open new possibilities for accurately modeling systems where unpredictability and discrete events are prevalent, such as financial markets, social behaviors, or sensor networks. Theoretically, it extends the applicability of neural differential equation models, offering a principled approach to include stochastic event handling within the continuous flow of systems.

Future Prospects

This research lays the groundwork for future exploration into more complex event-driving dynamics and their integration into broader AI systems. Potential directions include the incorporation of more sophisticated stochastic modeling techniques or the application of Neural JSDEs to emerging fields that require a hybrid approach to modeling. As computational power and modeling techniques evolve, Neural JSDEs could become integral to various predictive and prescriptive analytical frameworks.

In conclusion, this paper provides a significant contribution to the field of dynamical systems modeling. By effectively bridging continuous and discrete dynamics through a neural approach, it paves the way for a deeper understanding and more precise modeling of complex real-world systems.