Skip the Equations: Learning Behavior of Personalized Dynamical Systems Directly From Data

Krzysztof Kacprzyk, Julianna Piskorz, Mihaela van der Schaar

TL;DR: An interpretable framework for trajectory prediction from static features: a decision tree identifies the curve’s global shape, while generalized additive models assign the key properties such as start, peak, or inflection.

Abstract: While black-box approaches are commonly used for data-driven modeling of dynamical systems, they often obscure a system's underlying behavior and properties, limiting adoption in areas such as medicine and pharmacology. A two-step process of discovering ordinary differential equations (ODEs) and their subsequent mathematical analysis can yield insights into the system's dynamics. However, this analysis may be infeasible for complex equations, and refining the ODE to meet certain behavioral requirements can be challenging. Direct semantic modeling has recently been proposed to address these issues by predicting the system's behavior, such as the trajectory's shape, directly from data, bypassing post-hoc mathematical analysis. In this work, we extend the original instantiation, limited to one-dimensional trajectories and inputs, to accommodate multi-dimensional trajectories with additional personalization, allowing evolution to depend on auxiliary static features (e.g., patient covariates). In a series of experiments, we show how our approach enables practitioners to integrate prior knowledge, understand the dynamics, ensure desired behaviors, and revise the model when necessary.

Lay Summary: When we need to understand how something changes over time—be it tumor size, battery charge, or drug concentration—seeing the reason behind the forecast matters. Yet, most modern models remain sealed inside black boxes. We present a transparent learning framework that forecasts how any quantity evolves over time, clearly showing how each feature (e.g., age, weight, biomarkers) influences that curve. A compact decision tree first predicts the curve's overall shape (e.g., steadily rising or peaking, then falling). A simple additive model then fills in the details, such as the starting point, peak height, and long-term limit. By writing each of these values as a sum of simple one-feature curves, this means you can plot how, for instance, age impacts peak height, or temperature impacts the decay rate and read off every factor's individual effect. Because both parts are readable and editable, domain experts—from doctors to engineers—can inject prior knowledge, verify that the model adheres to known dynamics, and quickly correct any unwanted behavior. In contrast to previous work that only allowed for dependence on one variable (the starting point), we extend it to handle multiple variables simultaneously, enabling truly personalized trajectories.

Link To Code: https://github.com/krzysztof-kacprzyk/EPISODE

Primary Area: Applications->Chemistry, Physics, and Earth Sciences

Keywords: dynamical systems, differential equations, ODE discovery, interpretability, transparency, verification

Submission Number: 13671