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Breaking Free: Decoupling Forced Systems with Laplace Neural Networks

Bernd Zimmering, Cecília Coelho, Vaibhav Gupta, Maria Maleshkova, Oliver Niggemann

TL;DR

The paper addresses the challenge of modeling forced dynamical systems with external inputs and memory effects. It introduces LP-Net, a solver-free neural network that operates in the Laplace domain and explicitly decouples internal dynamics, initial conditions, and forcing. Key innovations include history-based encoding of initial states, a neural transfer function learned on a stabilized ILT grid, and an inverse-Laplace-transform reconstruction with time-scale stabilization. Across eight univariate benchmarks spanning linear, nonlinear, chaotic, and delayed dynamics, LP-Net consistently outperforms the Laplace Neural Operator and often surpasses LSTM, demonstrating improved accuracy, transferability, and interpretability for complex forcing scenarios.

Abstract

Modelling forced dynamical systems - where an external input drives the system state - is critical across diverse domains such as engineering, finance, and the natural sciences. In this work, we propose Laplace-Net, a decoupled, solver-free neural framework for learning forced and delay-aware systems. It leverages a Laplace transform-based approach to decompose internal dynamics, external inputs, and initial values into established theoretical concepts, enhancing interpretability. Laplace-Net promotes transferability since the system can be rapidly re-trained or fine-tuned for new forcing signals, providing flexibility in applications ranging from controller adaptation to long-horizon forecasting. Experimental results on eight benchmark datasets - including linear, non-linear, and delayed systems - demonstrate the method's improved accuracy and robustness compared to state-of-the-art approaches, particularly in handling complex and previously unseen inputs.

Breaking Free: Decoupling Forced Systems with Laplace Neural Networks

TL;DR

The paper addresses the challenge of modeling forced dynamical systems with external inputs and memory effects. It introduces LP-Net, a solver-free neural network that operates in the Laplace domain and explicitly decouples internal dynamics, initial conditions, and forcing. Key innovations include history-based encoding of initial states, a neural transfer function learned on a stabilized ILT grid, and an inverse-Laplace-transform reconstruction with time-scale stabilization. Across eight univariate benchmarks spanning linear, nonlinear, chaotic, and delayed dynamics, LP-Net consistently outperforms the Laplace Neural Operator and often surpasses LSTM, demonstrating improved accuracy, transferability, and interpretability for complex forcing scenarios.

Abstract

Modelling forced dynamical systems - where an external input drives the system state - is critical across diverse domains such as engineering, finance, and the natural sciences. In this work, we propose Laplace-Net, a decoupled, solver-free neural framework for learning forced and delay-aware systems. It leverages a Laplace transform-based approach to decompose internal dynamics, external inputs, and initial values into established theoretical concepts, enhancing interpretability. Laplace-Net promotes transferability since the system can be rapidly re-trained or fine-tuned for new forcing signals, providing flexibility in applications ranging from controller adaptation to long-horizon forecasting. Experimental results on eight benchmark datasets - including linear, non-linear, and delayed systems - demonstrate the method's improved accuracy and robustness compared to state-of-the-art approaches, particularly in handling complex and previously unseen inputs.

Paper Structure

This paper contains 28 sections, 34 equations, 2 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Neural ODEs and Solver-Based Extensions.
  4. Laplace-Based, Solver-Free Methods.
  5. Operator Learning for Forced Systems.
  6. Bridging Gaps via Decoupling and Generalized Forcing.
  7. Preliminaries
  8. The Laplace transform
  9. Solution
  10. Decomposition of System Responses
  11. Neural Network-based Approximation
  12. External Input $X(s)$:
  13. History Encoding and Initial State $\boldsymbol{\mathcal{P}}(s)$
  14. Transfer Function $\boldsymbol{\mathcal{H}}(s)$
  15. Inverse Laplace Transform and Time-Independent Scaling
  16. ...and 13 more sections

Figures (2)

  • Figure 1: Control loop with system $S$ that responds with $y(t)$ and is forced by some controller $C$ through excitations $x(t)$. $y_0$ is the initial state of the system.
  • Figure 2: Overview of the LP-Net architecture. Blue elements represent learnable matrices or NNs, including an encoder for historical data and a trainable transfer function. Purple elements denote complex-valued components.