Higher-Order LaSDI: Reduced Order Modeling with Multiple Time Derivatives

TL;DR

Higher-Order LaSDI (HLaSDI) presents a non-intrusive reduced-order modeling framework for parameterized PDEs with multiple time derivatives by employing K autoencoders linked through a shared, higher-order latent dynamical system. It introduces novel loss terms—Consistency, Chain-Rule, and Initial/Rollout variants—together with high-order finite-difference schemes on nonuniform time grids to enforce long-horizon accuracy. The method, built on and extending GPLaSDI and Rollout-LaSDI, demonstrates strong predictive performance across four challenging PDEs (Burger’s, Wave, Telegrapher’s, Klein-Gordon) with small training sets and substantial speedups (up to 40×) in inference time. These contributions broaden the applicability of ROMs to complex, multi-derivative PDEs, enabling efficient, reliable surrogate modeling for parameterized systems in scientific computing.

Abstract

Solving complex partial differential equations is vital in the physical sciences, but often requires computationally expensive numerical methods. Reduced-order models (ROMs) address this by exploiting dimensionality reduction to create fast approximations. While modern ROMs can solve parameterized families of PDEs, their predictive power degrades over long time horizons. We address this by (1) introducing a flexible, high-order, yet inexpensive finite-difference scheme and (2) proposing a Rollout loss that trains ROMs to make accurate predictions over arbitrary time horizons. We demonstrate our approach on the 2D Burgers equation.

Figures (14)

• Figure 1: A schematic for a parameterized ROM. A ROM can be used to predict the future FOM state by a) encoding the initial FOM state, b) using the latent dynamics (denoted here as $G$) to integrate the initial latent state, and c) decoding the final latent state to predict the future FOM state.
• Figure 2: A schematic of Rollout. From left to right, we begin by encoding a discretization $\vec{u}_{\theta}(t)$ of the FOM solution, $u_{\theta}(t, X)$. This gives us the initial latent state, $\vec{z}_{\theta}(t)$. We sample the posterior distribution at $\theta$ of the gaussian processes for the latent dynamics coefficients to obtain a set of latent coefficients for parameter $\theta$. We use these samples to define the latent coefficients, yielding a dynamical system which we can integrate forward to obtain a prediction of the future latent state, $\hat{z}_{\theta}\left(t + \Delta t^{\theta} \right)$. We decode this to obtain $\hat{u}_{\theta}\left(t + \Delta t^{\theta} \right)$, a prediction of the future FOM state.
• Figure 3: A schematic of Rollout. From left to right, we begin by encoding a discretization $\vec{u}_{\theta}(t)$ of the FOM solution, $u_{\theta}(t, X)$. This gives us the initial latent state, $\vec{z}_{\theta}(t)$. We use the latent dynamics for $\theta$ to integrate the initial latent state and obtain a prediction of the future latent state, $\hat{z}_{\theta}\left(t + \Delta t^{\theta} \right)$. We decode this to obtain $\hat{u}_{\theta}\left(t + \Delta t^{\theta} \right)$, a prediction of the future FOM state, which we compare to an interpolation $\tilde{u}_{\theta}\left( t + \Delta t^{\theta} \right)$ of the future FOM state.
• Figure 4: A schematic of HLaSDI's architecture, consisting of $K$ autoencoders with a unified set of latent dynamics. The $k$'th encoder, $\varphi_e^{(k)}$ encodes snapshots of the $k$'th time derivative of the FOM solution. Each Autoencoder maps to the same latent space, and we assume that their latent states are governed by a common set of latent dynamics, equation \ref{['eq:Latent:ODE:higher']}.
• Figure 5: Relative error of (left) displacement and (right) velocity applying HLaSDI to the 1D Burger's equation \ref{['eq:1dburg']}. Red squares represent initial training points and black squares represent training points chosen through greedy sampling during training. All other grid points are testing parameters. HLaSDI achieves errors under 8.1% using only 10 data points