Implicit Neural Spatial Representation (INSR) has emerged as an effective representation of spatially-dependent vector fields. This work explores solving time-dependent PDEs with INSR. Classical PDE solvers introduce both temporal and spatial discretizations. Common spatial discretizations include meshes and meshless point clouds, where each degree-of-freedom corresponds to a location in space. While these explicit spatial correspondences are intuitive to model and understand, these representations are not necessarily optimal for accuracy, memory usage, or adaptivity. Keeping the classical temporal discretization unchanged (e.g., explicit/implicit Euler), we explore INSR as an alternative spatial discretization, where spatial information is implicitly stored in the neural network weights. The network weights then evolve over time via time integration. Our approach does not require any training data generated by existing solvers because our approach is the solver itself. We validate our approach on various PDEs with examples involving large elastic deformations, turbulent fluids, and multi-scale phenomena. While slower to compute than traditional representations, our approach exhibits higher accuracy and lower memory consumption. Whereas classical solvers can dynamically adapt their spatial representation only by resorting to complex remeshing algorithms, our INSR approach is intrinsically adaptive. By tapping into the rich literature of classic time integrators, e.g., operator-splitting schemes, our method enables challenging simulations in contact mechanics and turbulent flows where previous neural-physics approaches struggle. Videos and codes are available on the project page: http://www.cs.columbia.edu/cg/INSR-PDE/

翻译：隐式神经空间表示（INSR）已成为一种有效的空间依赖向量场表示方法。本文探索了利用INSR求解含时偏微分方程（PDE）的途径。经典PDE求解器同时引入时间和空间离散化。常见的空间离散化方法包括网格和无网格点云，其中每个自由度对应空间中的一个位置。尽管这些显式空间对应关系在建模和理解上直观，但这些表示在精度、内存使用或自适应性方面并非最优。保持经典时间离散化不变（例如显式/隐式欧拉法），我们探索将INSR作为替代的空间离散化方法，其中空间信息隐式存储于神经网络权重中。网络权重随后通过时间积分随时间演化。我们的方法无需由现有求解器生成的任何训练数据，因为本方法本身就是求解器。我们通过涉及大弹性变形、湍流流体和多尺度现象的多种PDE算例验证了该方法。虽然计算速度慢于传统表示方法，但我们的方法展现出更高的精度和更低的内存消耗。经典求解器只能通过复杂的网格重构算法动态调整空间表示，而我们的INSR方法本质上具有自适应性。通过借鉴经典时间积分器（如算子分裂格式）的丰富文献，我们的方法得以实现接触力学和湍流流动等具有挑战性的仿真，而以往的神经物理方法在这些场景中难以胜任。相关视频和代码可在项目页面获取：http://www.cs.columbia.edu/cg/INSR-PDE/