Learning non-Markovian Dynamical Systems with Signature-based Encoders

2025-09-15

AI Summary

Modeling non-Markovian dynamical systems in continuous time—where evolution depends on the entire history rather than just the current state—remains challenging due to discretization artifacts and lack of theoretical guarantees in existing RNN-based approaches. Method: We propose the first neural differential equation framework grounded in the signature transform, using signatures as a continuous-time history encoder that intrinsically captures path-dependent memory and delay effects—overcoming the instantaneous-state limitation of standard Neural ODEs. The signature provides a differentiable, stable, and theoretically principled representation of historical trajectories. Contribution/Results: Integrated with neural ODEs, our framework forms an end-to-end continuous-time encoding-decoding model. On synthetic benchmarks, it significantly outperforms RNN baselines in both predictive accuracy and training stability, empirically validating the efficacy and superiority of signature-based representations for non-Markovian dynamics modeling.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Deep Generative Models & Autoencoders
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Neural ordinary differential equations offer an effective framework for modeling dynamical systems by learning a continuous-time vector field. However, they rely on the Markovian assumption - that future states depend only on the current state - which is often untrue in real-world scenarios where the dynamics may depend on the history of past states. This limitation becomes especially evident in settings involving the continuous control of complex systems with delays and memory effects. To capture historical dependencies, existing approaches often rely on recurrent neural network (RNN)-based encoders, which are inherently discrete and struggle with continuous modeling. In addition, they may exhibit poor training behavior. In this work, we investigate the use of the signature transform as an encoder for learning non-Markovian dynamics in a continuous-time setting. The signature transform offers a continuous-time alternative with strong theoretical foundations and proven efficiency in summarizing multidimensional information in time. We integrate a signature-based encoding scheme into encoder-decoder dynamics models and demonstrate that it outperforms RNN-based alternatives in test performance on synthetic benchmarks.

Problem

• Modeling non-Markovian dynamics with historical dependencies
• Overcoming limitations of recurrent neural network encoders
• Capturing continuous-time memory effects in complex systems

Innovation

• Signature transform encoder for non-Markovian dynamics
• Continuous-time alternative to RNN-based encoders
• Encoder-decoder models with signature-based encoding