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 **Ph**ysics-**i**nformed **ROM** ($\Phi$-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, $\Phi$-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.

Nonuniform grid refinement plays a fundamental role in simulating realistic flows with a multitude of length scales. We introduce the first GPU-optimized implementation of this technique in the context of the lattice Boltzmann method. Our approach focuses on enhancing GPU performance while minimizing memory access bottlenecks. We employ kernel fusion techniques to optimize memory access patterns, reduce synchronization overhead, and minimize kernel launch latencies. Additionally, our implementation ensures efficient memory management, resulting in lower memory requirements compared to the baseline LBM implementations that were designed for distributed systems. Our implementation allows simulations of unprecedented domain size (e.g., 1596 × 840 × 840) using a single A100-40 GB GPU thanks to enabling grid refinement capabilities on a single GPU. We validate our code against published experimental data. Our optimization improves the performance of the baseline algorithm by 1.3–2X. We also compare against state-of-the-art current solutions for grid refinement LBM and show an order of magnitude speedup.