Physics-informed Reduced Order Modeling of Time-dependent PDEs via Differentiable Solvers

TL;DR

Φ-ROM addresses the challenge of ensuring physical fidelity in reduced-order models for time-dependent PDEs by embedding the governing physics directly into the training of a nonlinear ROM. It uses a differentiable PDE solver to constrain latent-space dynamics through a physics-informed loss, linking the decoder and latent evolution so that $\dot{\alpha}=\Psi_{\phi}(\alpha)$ aligns with $\frac{d\hat{u}}{dt}$ produced by the solver. The framework supports mesh-free, irregular-grid data via conditional INRs, and employs hyper-reduction to make training scalable. Across diffusion, Burgers', Navier–Stokes, KdV, and LBM problems, Φ-ROM outperforms purely data-driven baselines and other physics-informed approaches, especially in generalization to unseen parameters, long-horizon forecasting, and sparse observations. The method advances real-time, physics-consistent simulations and data assimilation, with an open-source JAX implementation for extensibility to other PDE systems and differentiable solvers.

Abstract

Reduced-order modeling (ROM) of time-dependent and parameterized differential equations aims to accelerate the simulation of complex high-dimensional systems by learning a compact latent manifold representation that captures the characteristics of the solution fields and their time-dependent dynamics. Although high-fidelity numerical solvers generate the training datasets, they have thus far been excluded from the training process, causing the learned latent dynamics to drift away from the discretized governing physics. This mismatch often limits generalization and forecasting capabilities. In this work, we propose Physics-informed ROM ($Φ$-ROM) by incorporating differentiable PDE solvers into the training procedure. Specifically, the latent space dynamics and its dependence on PDE parameters are shaped directly by the governing physics encoded in the solver, ensuring a strong correspondence between the full and reduced systems. Our model outperforms state-of-the-art data-driven ROMs and other physics-informed strategies by accurately generalizing to new dynamics arising from unseen parameters, enabling long-term forecasting beyond the training horizon, maintaining continuity in both time and space, and reducing the data cost. Furthermore, $Φ$-ROM learns to recover and forecast the solution fields even when trained or evaluated with sparse and irregular observations of the fields, providing a flexible framework for field reconstruction and data assimilation. We demonstrate the framework's robustness across various PDE solvers and highlight its broad applicability by providing an open-source JAX implementation that is readily extensible to other PDE systems and differentiable solvers, available at https://phi-rom.github.io.

Figures (13)

• Figure 1: Inference with $\Phi$-ROM. The initial condition and the forecast solution at a target time $T_*$ can be observed and reconstructed on arbitrary (and possibly different) grids $\mathcal{X}$ and $\mathcal{X}'$.
• Figure 2: (Left) Training $\Phi$-ROM with the reconstruction loss and dynamics loss. Red and purple dashed lines show the gradient path taken by $L_{rec}$ and $L_{dyn}$. The reconstruction loss trains $D_\theta$ and $\alpha$ with spatial information, while the dynamics loss trains the $\Psi_\phi$ and regularizes the decoder and solution manifolds $\Gamma$ with physics information from $\mathcal{S}$. (Right) $\Phi$-ROM can be trained with partial and irregular observations of the solution fields and recover the full fields and dynamics.
• Figure 3: Forecasting Burgers' (left) and N-S (right) test samples.
• Figure 4: RNMSE for N-S based on $\Phi$-ROM and DINo for $\left[0, T_{tr}\right]$ and $\left[T_{tr}, T_{te}\right]$ with increasing size of the training dataset, showing data efficiency.
• Figure 5: $D_\theta$ reconstructs the solution field one coordinate at a time and stacks the solution at all coordinates for the full field.

Theorems & Definitions (3)

• Remark 2.1
• Remark 3.1
• Remark 3.2