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No Equations Needed: Learning System Dynamics Without Relying on Closed-Form ODEs

Krzysztof Kacprzyk, Mihaela van der Schaar

TL;DR

This work introduces direct semantic modeling as an alternative to the traditional two-step process of discovering closed-form ODEs and then analyzing their behavior. By predicting the semantic representation (composition and property sets) directly from data through a semantic predictor and then obtaining trajectories via a trajectory predictor, the approach bypasses post-hoc mathematical analysis and enables intuitive editing and semantic priors. The authors formalize semantic representations using motif-based compositions with bounded and unbounded dynamics, and instantiate a concrete model called Semantic ODE for one-dimensional systems. Empirical evidence in pharmacokinetics, tumor growth, and other synthetic datasets shows competitive or superior performance to standard ODE-discovery methods while offering greater interpretability, robustness to noise, and easy constraint editing. The framework lays out a path toward more transparent and flexible dynamical-system modeling, with clear extensions to multi-dimensional and more complex dynamics in future work.

Abstract

Data-driven modeling of dynamical systems is a crucial area of machine learning. In many scenarios, a thorough understanding of the model's behavior becomes essential for practical applications. For instance, understanding the behavior of a pharmacokinetic model, constructed as part of drug development, may allow us to both verify its biological plausibility (e.g., the drug concentration curve is non-negative and decays to zero) and to design dosing guidelines. Discovery of closed-form ordinary differential equations (ODEs) can be employed to obtain such insights by finding a compact mathematical equation and then analyzing it (a two-step approach). However, its widespread use is currently hindered because the analysis process may be time-consuming, requiring substantial mathematical expertise, or even impossible if the equation is too complex. Moreover, if the found equation's behavior does not satisfy the requirements, editing it or influencing the discovery algorithms to rectify it is challenging as the link between the symbolic form of an ODE and its behavior can be elusive. This paper proposes a conceptual shift to modeling low-dimensional dynamical systems by departing from the traditional two-step modeling process. Instead of first discovering a closed-form equation and then analyzing it, our approach, direct semantic modeling, predicts the semantic representation of the dynamical system (i.e., description of its behavior) directly from data, bypassing the need for complex post-hoc analysis. This direct approach also allows the incorporation of intuitive inductive biases into the optimization algorithm and editing the model's behavior directly, ensuring that the model meets the desired specifications. Our approach not only simplifies the modeling pipeline but also enhances the transparency and flexibility of the resulting models compared to traditional closed-form ODEs.

No Equations Needed: Learning System Dynamics Without Relying on Closed-Form ODEs

TL;DR

This work introduces direct semantic modeling as an alternative to the traditional two-step process of discovering closed-form ODEs and then analyzing their behavior. By predicting the semantic representation (composition and property sets) directly from data through a semantic predictor and then obtaining trajectories via a trajectory predictor, the approach bypasses post-hoc mathematical analysis and enables intuitive editing and semantic priors. The authors formalize semantic representations using motif-based compositions with bounded and unbounded dynamics, and instantiate a concrete model called Semantic ODE for one-dimensional systems. Empirical evidence in pharmacokinetics, tumor growth, and other synthetic datasets shows competitive or superior performance to standard ODE-discovery methods while offering greater interpretability, robustness to noise, and easy constraint editing. The framework lays out a path toward more transparent and flexible dynamical-system modeling, with clear extensions to multi-dimensional and more complex dynamics in future work.

Abstract

Data-driven modeling of dynamical systems is a crucial area of machine learning. In many scenarios, a thorough understanding of the model's behavior becomes essential for practical applications. For instance, understanding the behavior of a pharmacokinetic model, constructed as part of drug development, may allow us to both verify its biological plausibility (e.g., the drug concentration curve is non-negative and decays to zero) and to design dosing guidelines. Discovery of closed-form ordinary differential equations (ODEs) can be employed to obtain such insights by finding a compact mathematical equation and then analyzing it (a two-step approach). However, its widespread use is currently hindered because the analysis process may be time-consuming, requiring substantial mathematical expertise, or even impossible if the equation is too complex. Moreover, if the found equation's behavior does not satisfy the requirements, editing it or influencing the discovery algorithms to rectify it is challenging as the link between the symbolic form of an ODE and its behavior can be elusive. This paper proposes a conceptual shift to modeling low-dimensional dynamical systems by departing from the traditional two-step modeling process. Instead of first discovering a closed-form equation and then analyzing it, our approach, direct semantic modeling, predicts the semantic representation of the dynamical system (i.e., description of its behavior) directly from data, bypassing the need for complex post-hoc analysis. This direct approach also allows the incorporation of intuitive inductive biases into the optimization algorithm and editing the model's behavior directly, ensuring that the model meets the desired specifications. Our approach not only simplifies the modeling pipeline but also enhances the transparency and flexibility of the resulting models compared to traditional closed-form ODEs.

Paper Structure

This paper contains 116 sections, 5 theorems, 78 equations, 16 figures, 15 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background: data-driven modeling of dynamical systems through ODE discovery.
  3. Motivation: the primary goal of discovering a closed-form ODE is its semantic representation.
  4. Limitations of the traditional two-step modeling.
  5. Proposed approach: direct semantic modeling.
  6. Contributions and outline.
  7. Forecasting models and discovery of closed-form ODEs
  8. From discovery and analysis to direct semantic modeling
  9. Syntax vs. semantics.
  10. Two-step modeling and its limitations
  11. Direct semantic modeling
  12. Forecasting model determined by semantic representation
  13. Semantic ODE as a concrete instantiation
  14. Formalizing semantic representation
  15. Semantic representation as composition and properties
  16. ...and 101 more sections

Key Result

Theorem 1

Let $u_|,v_|$ be bounded trajectory parts on $(t_0,t_{\text{end}})$ and assume $v_| \equiv (c_|,p_|)$, where $(c,p)$ is a semantic representation. If both of the following hold for every pair of consecutive transition points ($a,b$) of $v_|$ then $u_| \sim (c_|,p_|) \implies u_| \equiv (c_|,p_|)$.

Figures (16)

  • Figure 1: Syntactic representation of a logistic growth model refers to its symbolic form, whereas semantic representation describes its behavior for different initial conditions.
  • Figure 2: Comparison between two-step modeling and direct semantic modeling. Left: The discovery of closed-form ODEs often allows for human analysis, but editing the equation or providing feedback to the optimization algorithm is challenging. Right: We propose to predict the semantic representation directly from data, which allows for editing the model and steering the optimization algorithm.
  • Figure 3: (a) Composition and transition points of $x(t)=\sin(t)$ on $[0,2\pi]$. (b) Motif set used in the proposed formalization of semantic representation.
  • Figure 4: Semantic ODE ($F$) maps the initial condition to the semantic representation of the trajectory (using $F_{\text{sem}}$ and then uses it to predict the actual trajectory (through $F_{\text{traj}}$). Formally, $F=F_{\text{traj}} \circ F_{\text{sem}}$.
  • Figure 5: Property sub-map of the logistic growth model (for composition $(s_{++b},s_{+-h})$) describes how various properties depend on the initial condition. See \ref{['fig:semantic_representation_logistic']} for a full semantic representation.
  • ...and 11 more figures

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • proof
  • ...and 6 more