Rederived Closed-Form Continuous-Time Neural Networks

In the field of machine learning, continuous-time neural networks (CTNNs) have garnered significant attention due to their powerful modeling capabilities for time series data. However, traditional numerical ordinary differential equation (ODE) solving methods suffer from computational inefficiency. Although closed-form CTNNs (CfCs) alleviate this issue through closed-form approximate solutions, their zero-order approximation method based on segmented constant inputs still exhibits errors when processing high-resolution or irregularly sampled data. This study proposes a numerical integration approximation theory based on the Lagrange interpolation method (NIALIM), which introduces temporal interval factors to linearly approximate the nonlinear integral function in liquid time-constant networks (LTCs) network state equations, deriving closed-form solutions dependent on external inputs and sampling intervals, while rigorously proving the error upper bound between the approximate and original solutions. Furthermore, we construct two novel network architectures: the dynamic feature accumulation closed-form network (DFA-CfN) for general sequence tasks, and the prereconstructed DFA-CfN (PRDFA-CfN) optimized for decomposable sequence tasks, along with six variant networks from these two families. Extensive experiments across six distinct tasks-physical dynamics modeling, human activity recognition, PhysioNet challenge, internet movie database (IMDb) sentiment analysis, XCO₂ remote sensing sequence prediction, and multienergy load forecasting in integrated energy systems (IESs)-demonstrate that our proposed models achieve state-of-the-art performance across all benchmarks, with the highest metrics improvement reaching 69.2%, providing a novel high-performance tool for continuous-time modeling tasks. The code used to support the findings of this study is available from https://github.com/nuist-xf/DFA-CfNs.