The long runtime of high-fidelity partial differential equation (PDE) solvers makes them unsuitable for time-critical applications. We propose to accelerate PDE solvers using reduced-order modeling (ROM). Whereas prior ROM approaches reduce the dimensionality of discretized vector fields, our continuous reduced-order modeling (CROM) approach builds a low-dimensional embedding of the continuous vector fields themselves, not their discretization. We represent this reduced manifold using continuously differentiable neural fields, which may train on any and all available numerical solutions of the continuous system, even when they are obtained using diverse methods or discretizations. We validate our approach on an extensive range of PDEs with training data from voxel grids, meshes, and point clouds. Compared to prior discretization-dependent ROM methods, such as linear subspace proper orthogonal decomposition (POD) and nonlinear manifold neural-network-based autoencoders, CROM features higher accuracy, lower memory consumption, dynamically adaptive resolutions, and applicability to any discretization. For equal latent space dimension, CROM exhibits 79$\times$ and 49$\times$ better accuracy, and 39$\times$ and 132$\times$ smaller memory footprint, than POD and autoencoder methods, respectively. Experiments demonstrate 109$\times$ and 89$\times$ wall-clock speedups over unreduced models on CPUs and GPUs, respectively. Videos and codes are available on the project page: https://crom-pde.github.io

翻译：高保真偏微分方程（PDE）求解器的长运行时间使其不适用于时间关键型应用。我们提出使用降阶建模（ROM）来加速PDE求解器。先前的ROM方法降低离散化向量场的维度，而我们的连续降阶建模（CROM）方法则构建连续向量场本身的低维嵌入，而非其离散化。我们使用连续可微神经场来表示这个降维流形，该神经场可在连续系统的任何及所有可用数值解上进行训练，即使这些解是通过不同方法或离散化方式获得的。我们在涵盖体素网格、网格和点云训练数据的广泛PDE上验证了该方法。与先前依赖于离散化的ROM方法（如线性子空间本征正交分解（POD）和基于非线性流形神经网络的自编码器）相比，CROM具有更高精度、更低内存消耗、动态自适应分辨率以及适用于任何离散化的特性。在相同的潜在空间维度下，与POD和自编码器方法相比，CROM的精度分别提升79倍和49倍，内存占用分别减少39倍和132倍。实验表明，与CPU和GPU上的未降阶模型相比，实际运行速度分别提升109倍和89倍。视频和代码可在项目页面获取：https://crom-pde.github.io