Neural Network-Based Parameter Estimation for Non-Autonomous Differential Equations with Discontinuous Signals

2025-07-07

AI Summary

Parameter estimation for non-autonomous differential equations driven by discontinuous external signals remains challenging—conventional methods suffer from inaccuracy at signal discontinuities and poor fitting stability. Method: This paper proposes a neural network–driven, multi-stage joint optimization framework. Its core innovation lies in employing a neural network to learn a differentiable harmonic smoothing approximation of the discontinuous input signal, which is then jointly optimized with the differential equation parameters in an iterative manner, enabling synergistic improvement of signal reconstruction and dynamical modeling. Contribution/Results: The method significantly enhances model fidelity and robustness when fitting real-world data featuring abrupt changes—e.g., light-onset stimuli or pulsatile hormonal inputs. Validation on biological systems—including circadian oscillators and yeast mating response pathways—demonstrates superior parameter estimation accuracy and stability compared to state-of-the-art approaches, thereby expanding the class of non-autonomous dynamical models that can be reliably inferred from experimental measurements.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: State Estimation
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

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• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Economics, Online Markets and Human Computation: Social networks and social learning

Abstract

Non-autonomous differential equations are crucial for modeling systems influenced by external signals, yet fitting these models to data becomes particularly challenging when the signals change abruptly. To address this problem, we propose a novel parameter estimation method utilizing functional approximations with artificial neural networks. Our approach, termed Harmonic Approximation of Discontinuous External Signals using Neural Networks (HADES-NN), operates in two iterated stages. In the first stage, the algorithm employs a neural network to approximate the discontinuous signal with a smooth function. In the second stage, it uses this smooth approximate signal to estimate model parameters. HADES-NN gives highly accurate and precise parameter estimates across various applications, including circadian clock systems regulated by external light inputs measured via wearable devices and the mating response of yeast to external pheromone signals. HADES-NN greatly extends the range of model systems that can be fit to real-world measurements.

Problem

• Estimating parameters in non-autonomous differential equations with discontinuous signals
• Approximating discontinuous signals using neural networks for smoother functions
• Enhancing parameter estimation accuracy in systems with abrupt external changes

Innovation

• Neural network approximates discontinuous signals smoothly
• Two-stage method for parameter estimation
• Accurate parameter estimates for non-autonomous differential equations