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

2025-01-30

AI Summary

Conventional dynamical system modeling relies on complex closed-form ordinary differential equations (ODEs), suffering from poor interpretability, limited editability, and difficulty in enforcing prior behavioral constraints (e.g., non-negativity, asymptotic decay). Method: We propose “direct semantic modeling”—a novel paradigm that bypasses symbolic equation derivation and instead learns semantic representations of system behavior directly from data. Behavioral specifications (e.g., physical or biological plausibility) are encoded as differentiable optimization objectives within a neural ODE framework, supported by behavior-feature parameterization, physics-informed end-to-end training, and a differentiable behavioral verification module. Contribution/Results: Evaluated on low-dimensional systems such as pharmacokinetics, our approach achieves 98.7% behavioral compliance, improves debugging efficiency by 5×, and substantially reduces dependence on mathematical domain experts. It is the first method enabling behavior-level editability, formal verifiability, and high-fidelity dynamic modeling.

Technology Category

• Cognitive Modeling & Cognitive Systems: Simulating Human Behavior
• Search and Optimization: Sampling/Simulation-based Search
• Humans and AI: Human-Aware Planning and Behavior Prediction

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Dynamic System Modeling
• Complex Mathematical Equations
• Model Behavior Understanding

Innovation

• Data-driven modeling
• Model transparency
• Flexible optimization