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Machine Learning for highly oscillatory differential equations

Maxime Bouchereau

TL;DR

This work tackles the challenge of solving highly oscillatory differential equations by marrying averaging theory with neural networks to learn the slow-fast decomposition. It then leverages a micro-macro formulation to obtain uniformly accurate solutions across the oscillation parameter $\varepsilon\in(0,1]$, using neural surrogates for the averaged field and the high-oscillation generator, together with an auto-encoder (or an autonomous variant) to preserve structure. Theoretical error analyses show exponential remainders for slow-fast and UA bounds for micro-macro, supported by numerical experiments on inverted pendulum and Van der Pol systems that demonstrate improved accuracy and uniform reliability. The approach reduces pre-computation costs, scales to low-dimensional multiscale problems, and provides a practical, data-driven framework for efficient multiscale ODE integration.

Abstract

Highly oscillatory differential equations, commonly encountered in multi-scale problems, are often too complex to solve analytically. However, several numerical methods have been developed to approximate their solutions. Although these methods have shown their efficiency, the first part of the strategy often involves heavy pre-computations from averaging theory. In this paper, we leverage neural networks (machine learning) to approximate the vector fields required by the pre-computations in the first part, and combine this with micro-macro techniques to efficiently solve the oscillatory problem. We illustrate our work by numerical simulations.

Machine Learning for highly oscillatory differential equations

TL;DR

This work tackles the challenge of solving highly oscillatory differential equations by marrying averaging theory with neural networks to learn the slow-fast decomposition. It then leverages a micro-macro formulation to obtain uniformly accurate solutions across the oscillation parameter , using neural surrogates for the averaged field and the high-oscillation generator, together with an auto-encoder (or an autonomous variant) to preserve structure. Theoretical error analyses show exponential remainders for slow-fast and UA bounds for micro-macro, supported by numerical experiments on inverted pendulum and Van der Pol systems that demonstrate improved accuracy and uniform reliability. The approach reduces pre-computation costs, scales to low-dimensional multiscale problems, and provides a practical, data-driven framework for efficient multiscale ODE integration.

Abstract

Highly oscillatory differential equations, commonly encountered in multi-scale problems, are often too complex to solve analytically. However, several numerical methods have been developed to approximate their solutions. Although these methods have shown their efficiency, the first part of the strategy often involves heavy pre-computations from averaging theory. In this paper, we leverage neural networks (machine learning) to approximate the vector fields required by the pre-computations in the first part, and combine this with micro-macro techniques to efficiently solve the oscillatory problem. We illustrate our work by numerical simulations.
Paper Structure (36 sections, 3 theorems, 36 equations, 33 figures)

This paper contains 36 sections, 3 theorems, 36 equations, 33 figures.

Table of Contents

  1. Introduction
  2. Scope of the paper
  3. Related work
  4. Approximate solutions of highly oscillatory differential equations with machine learning
  5. General strategy
  6. Slow-fast decomposition
  7. Micro-Macro decomposition
  8. Machine learning method
  9. An alternative method for autonomous systems
  10. Error analysis
  11. Slow-fast decomposition
  12. Micro-Macro correction
  13. Alternative method
  14. Numerical experiments
  15. Approximation of the averaged field and high oscillation generator
  16. ...and 21 more sections

Key Result

Theorem 1

Let us denote the following learning errors: Let us consider these two hypotheses: Let $y_{\theta,n}^\varepsilon$ denote the following numerical flow: Then there exist constants $\lambda , \alpha > 0$ (independent of $h,\varepsilon)$ s.t. for all $h \leqslant h_+$ and $\varepsilon \in ]0,\varepsilon_0]$,

Figures (33)

  • Figure 1: Inverted pendulum with forward Euler method. Left: Error between $F^{[k]}$ and $F_{\theta}(\cdot,0,\varepsilon)$ for $k = 0,1$ w.r.t. $\varepsilon$. Right: Error between $\phi^{[k]}$ and $\phi_{\theta,+}(\cdot,\cdot,\varepsilon)$ for $k = 0,1$ w.r.t. $\varepsilon$.
  • Figure 2: Inverted pendulum with midpoint method. Left: Error between $F^{[k]}$ and $F_{\theta}(\cdot,0,\varepsilon)$ for $k = 0,1$ w.r.t. $\varepsilon$. Right: Error between $\phi^{[k]}$ and $\phi_{\theta,+}(\cdot,\cdot,\varepsilon)$ for $k = 0,1$ w.r.t. $\varepsilon$.
  • Figure 3: Van der Pol oscillator with forward Euler method. Left: Error between $F^{[k]}$ and $F_{\theta}(\cdot,0,\varepsilon)$ for $k = 0,1$ w.r.t. $\varepsilon$. Right: Error between $\phi^{[k]}$ and $\phi_{\theta,+}(\cdot,\cdot,\varepsilon)$ for $k = 0,1$ w.r.t. $\varepsilon$.
  • Figure 4: Comparison between $Loss$ decays (green: $Loss_{Train}$, red: $Loss_{Test}$), trajectories (dashed dark: exact flow, green: numerical flow with learned vector fields and local error (yellow) for the inverted pendulum with Forward Euler method in the case $\varepsilon = 0.001$.
  • Figure 5: Comparison between $Loss$ decays (green: $Loss_{Train}$, red: $Loss_{Test}$), trajectories (dashed dark: exact flow, green: numerical flow with learned vector fields and local error (yellow) for the inverted pendulum with Forward Euler method in the case $\varepsilon = 0.05$.
  • ...and 28 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3