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Using Neural Implicit Flow To Represent Latent Dynamics Of Canonical Systems

Imran Nasim, Joaõ Lucas de Sousa Almeida

TL;DR

The paper investigates learning latent dynamics of canonical PDEs using Neural Implicit Flow (NIF), a mesh-agnostic neural operator designed to encode latent states of dynamical systems. NIF is compared with the DeepONet family on three representative equations—fKdV, KS, and SG—treating NIF as a dimensionality-reduction mechanism via two subnetworks (ParameterNet and ShapeNet) that project data into a compact latent space. Experiments on the forced Korteweg–de Vries, Kuramoto–Sivashinsky, and Sine–Gordon equations show that NIF achieves lower reconstruction errors than DeepONet (e.g., KS bursting $1.68\%$ vs $12.3\%$, $2.0\%$ vs $5.0\%$, $2.68\%$ vs $7.48\%$) and reveals latent dynamics that track phase transitions, though its latent space is less interpretable than Fourier-like DeepONet representations. The results indicate that NIF provides a powerful low-dimensional surrogate for dynamical data with potential applications in reduced-order modelling and data-driven analysis, while highlighting opportunities to enhance latent interpretability through regularization and architecture design.

Abstract

The recently introduced class of architectures known as Neural Operators has emerged as highly versatile tools applicable to a wide range of tasks in the field of Scientific Machine Learning (SciML), including data representation and forecasting. In this study, we investigate the capabilities of Neural Implicit Flow (NIF), a recently developed mesh-agnostic neural operator, for representing the latent dynamics of canonical systems such as the Kuramoto-Sivashinsky (KS), forced Korteweg-de Vries (fKdV), and Sine-Gordon (SG) equations, as well as for extracting dynamically relevant information from them. Finally we assess the applicability of NIF as a dimensionality reduction algorithm and conduct a comparative analysis with another widely recognized family of neural operators, known as Deep Operator Networks (DeepONets).

Using Neural Implicit Flow To Represent Latent Dynamics Of Canonical Systems

TL;DR

The paper investigates learning latent dynamics of canonical PDEs using Neural Implicit Flow (NIF), a mesh-agnostic neural operator designed to encode latent states of dynamical systems. NIF is compared with the DeepONet family on three representative equations—fKdV, KS, and SG—treating NIF as a dimensionality-reduction mechanism via two subnetworks (ParameterNet and ShapeNet) that project data into a compact latent space. Experiments on the forced Korteweg–de Vries, Kuramoto–Sivashinsky, and Sine–Gordon equations show that NIF achieves lower reconstruction errors than DeepONet (e.g., KS bursting vs , vs , vs ) and reveals latent dynamics that track phase transitions, though its latent space is less interpretable than Fourier-like DeepONet representations. The results indicate that NIF provides a powerful low-dimensional surrogate for dynamical data with potential applications in reduced-order modelling and data-driven analysis, while highlighting opportunities to enhance latent interpretability through regularization and architecture design.

Abstract

The recently introduced class of architectures known as Neural Operators has emerged as highly versatile tools applicable to a wide range of tasks in the field of Scientific Machine Learning (SciML), including data representation and forecasting. In this study, we investigate the capabilities of Neural Implicit Flow (NIF), a recently developed mesh-agnostic neural operator, for representing the latent dynamics of canonical systems such as the Kuramoto-Sivashinsky (KS), forced Korteweg-de Vries (fKdV), and Sine-Gordon (SG) equations, as well as for extracting dynamically relevant information from them. Finally we assess the applicability of NIF as a dimensionality reduction algorithm and conduct a comparative analysis with another widely recognized family of neural operators, known as Deep Operator Networks (DeepONets).
Paper Structure (5 sections, 5 figures)

This paper contains 5 sections, 5 figures.

Table of Contents

  1. Introduction
  2. Models and Experiments
  3. Results
  4. Impact statement
  5. Results for Sine-Gordon Equations

Figures (5)

  • Figure 1: NIF and DeepONet architectures.
  • Figure 2: Latent variable profile with the pointwise error and predicted dynamics using NIF (upper panels) and DeepONet (lower panels).
  • Figure 3: The three-dimensional latent dynamics for KS (top) and fKdV (bottom) bursting dynamics data produced by i) Fourier projection (left panel) ii) DeepONet (middle panel) and iii) NIF (right panel).
  • Figure 4: Latent variable profile with the pointwise error and predicted dynamics using i) DeepONet (left panel) and ii) NIF (right panel).
  • Figure 5: The three-dimensional latent dynamics for Sine-Gordon bursting dynamics data produced by i) Fourier projection (left panel) and ii) DeepONet (middle panel) and iii) NIF (right panel).