ICML 2025

Skip the Equations: Learning Behavior of Personalized Dynamical Systems
Directly From Data
Krzysztof Kacprzyk 1 Julianna Piskorz 1 Mihaela van der Schaar 1
Abstract
While black-box approaches are commonly used
for data-driven modeling of dynamical systems,
they often obscure a system’s underlying behav-
ior and properties, limiting adoption in areas such
as medicine and pharmacology. A two-step pro-
cess of discovering ordinary differential equations
(ODEs) and their subsequent mathematical anal-
ysis can yield insights into the system’s dynam-
ics. However, this analysis may be infeasible for
complex equations, and refining the ODE to meet
certain behavioral requirements can be challeng-
ing. 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 mathe-
matical analysis. In this work, we extend the
original instantiation, limited to one-dimensional
trajectories and inputs, to accommodate multi-
dimensional trajectories with additional personal-
ization, allowing evolution to depend on auxiliary
static features (e.g., patient covariates). In a se-
ries of experiments, we show how our approach
enables practitioners to integrate prior knowledge,
understand the dynamics, ensure desired behav-
iors, and revise the model when necessary.
1. Introduction
Personalized Dynamical Systems
In machine learning
(ML), modeling dynamical systems is a core activity with
extensive applications in fields such as physics (Rudy et al.,
2017), biology (Chen et al., 2019), engineering (Brunton
& Kutz, 2022), and medicine (Lee et al., 2020). We call
a dynamical system personalized when its evolution func-
tion depends on certain observable features. It is similar
to dynamical systems that depend on a parameter (as stud-
1University of Cambridge, Cambridge, UK. Correspondence
to: Krzysztof Kacprzyk <kk751@cam.ac.uk>.
Proceedings of the 42 nd International Conferen...