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PAINT: Parallel-in-time Neural Twins for Dynamical System Reconstruction

Andreas Radler, Vincent Seyfried, Stefan Pirker, Johannes Brandstetter, Thomas Lichtenegger

TL;DR

This work addresses the challenge of creating neural twins that can adapt at test time using real-time measurements. It introduces PAINT, a framework that learns a joint conditional distribution of states given measurements and performs state reconstruction in a sliding-window, parallel-in-time fashion, avoiding reliance on a fixed initial condition. The authors provide theoretical guarantees that PAINT is on-trajectory under a measurement-decay assumption and contrast this with autoregressive models that can drift, illustrating the phenomenon of over-reliance on the autoregressive state. They instantiate PAINT as FlowPAINT with Flow Matching and validate it on a challenging 2D turbulent jet dataset, demonstrating improved physical coherence, lower error growth, and meaningful uncertainty estimates relative to autoregressive baselines. The work advances neural twin design by offering an architecture-agnostic, interpretable approach with uncertainty quantification, suitable for real-time state estimation and decision-making in complex dynamical systems, while acknowledging high computational costs and proposing avenues for future improvement.

Abstract

Neural surrogates have shown great potential in simulating dynamical systems, while offering real-time capabilities. We envision Neural Twins as a progression of neural surrogates, aiming to create digital replicas of real systems. A neural twin consumes measurements at test time to update its state, thereby enabling context-specific decision-making. A critical property of neural twins is their ability to remain on-trajectory, i.e., to stay close to the true system state over time. We introduce Parallel-in-time Neural Twins (PAINT), an architecture-agnostic family of methods for modeling dynamical systems from measurements. PAINT trains a generative neural network to model the distribution of states parallel over time. At test time, states are predicted from measurements in a sliding window fashion. Our theoretical analysis shows that PAINT is on-trajectory, whereas autoregressive models generally are not. Empirically, we evaluate our method on a challenging two-dimensional turbulent fluid dynamics problem. The results demonstrate that PAINT stays on-trajectory and predicts system states from sparse measurements with high fidelity. These findings underscore PAINT's potential for developing neural twins that stay on-trajectory, enabling more accurate state estimation and decision-making.

PAINT: Parallel-in-time Neural Twins for Dynamical System Reconstruction

TL;DR

This work addresses the challenge of creating neural twins that can adapt at test time using real-time measurements. It introduces PAINT, a framework that learns a joint conditional distribution of states given measurements and performs state reconstruction in a sliding-window, parallel-in-time fashion, avoiding reliance on a fixed initial condition. The authors provide theoretical guarantees that PAINT is on-trajectory under a measurement-decay assumption and contrast this with autoregressive models that can drift, illustrating the phenomenon of over-reliance on the autoregressive state. They instantiate PAINT as FlowPAINT with Flow Matching and validate it on a challenging 2D turbulent jet dataset, demonstrating improved physical coherence, lower error growth, and meaningful uncertainty estimates relative to autoregressive baselines. The work advances neural twin design by offering an architecture-agnostic, interpretable approach with uncertainty quantification, suitable for real-time state estimation and decision-making in complex dynamical systems, while acknowledging high computational costs and proposing avenues for future improvement.

Abstract

Neural surrogates have shown great potential in simulating dynamical systems, while offering real-time capabilities. We envision Neural Twins as a progression of neural surrogates, aiming to create digital replicas of real systems. A neural twin consumes measurements at test time to update its state, thereby enabling context-specific decision-making. A critical property of neural twins is their ability to remain on-trajectory, i.e., to stay close to the true system state over time. We introduce Parallel-in-time Neural Twins (PAINT), an architecture-agnostic family of methods for modeling dynamical systems from measurements. PAINT trains a generative neural network to model the distribution of states parallel over time. At test time, states are predicted from measurements in a sliding window fashion. Our theoretical analysis shows that PAINT is on-trajectory, whereas autoregressive models generally are not. Empirically, we evaluate our method on a challenging two-dimensional turbulent fluid dynamics problem. The results demonstrate that PAINT stays on-trajectory and predicts system states from sparse measurements with high fidelity. These findings underscore PAINT's potential for developing neural twins that stay on-trajectory, enabling more accurate state estimation and decision-making.
Paper Structure (57 sections, 14 equations, 16 figures, 4 tables)

This paper contains 57 sections, 14 equations, 16 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Notation.
  4. Digital twins and data integration
  5. Digital twins
  6. Bayesian filtering.
  7. Reconstructing flows from sparse measurements.
  8. Scalable Neural Surrogates
  9. Surrogates models.
  10. Transformers.
  11. Generative modeling of PDEs.
  12. Computational Fluid Dynamics
  13. Incompressible Navier-Stokes.
  14. Precision–computation tradeoff.
  15. Method
  16. ...and 42 more sections

Figures (16)

  • Figure 1: Real world measurements allow for parallel-in-time models to stay on-trajectory, while autoregressive models may drift over time. Dashed lines represent sampled trajectories. Parallel-in-time models take a window of measurements and predict a distribution of subtrajectories which can be sampled and used for state and uncertainty estimation. Autoregressive models are prone to over-reliance on their autoregressive state (see Section \ref{['sec:theory']}) and typically depend on an initial state.
  • Figure 2: Interaction between a real-world dynamical system, simulations, and a neural twin. The workflow is structured in three phases: (1) The dynamical systems informs the design of the simulations. (2) Via the simulations, a diverse training dataset is generated, which is used for training the neural twin. (3) During inference, the neural twin integrates real-time measurement data to remain on-trajectory. (4) The state estimation of the neural twin can optionally be used for decision-making.
  • Figure 3: Probe positions.
  • Figure 4: Ground truth vs. predicted states at different timesteps for $Re = 2100$ using the vertical probe constellation. PAINT (ours) does not accumulate errors over time. For the autoregressive UNet a pronounced drift over time is observable. All values are normalized by the mean inlet velocity.
  • Figure 5: Mean velocity $\overline{u}$ and variance $\overline{u'^2}$ of the reconstructed $Re=2100$ test trajectory using the vertical probe constellation compared with the ground truth. All values are normalized by the mean inlet velocity.
  • ...and 11 more figures

Theorems & Definitions (1)

  • Definition 1