  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/
