We present ContinuousCloth, a continuous-time, physics-informed neural solver for deformable cloth dynamics that enforces a spring--mass ODE within a neural trajectory model. Unlike classical time-stepping simulators that return trajectories only on a discrete integration grid, ContinuousCloth supports querying at arbitrary timestamps and computing displacement, velocity, and acceleration via automatic differentiation. We evaluate ContinuousCloth along two dimensions. In a physics-only setting, ContinuousCloth achieves trajectory accuracy comparable to high-precision solvers across six diverse boundary-condition scenarios while reducing runtime by up to 17.4 times. When fused with sparse and noisy observations, the spring--mass ODE acts as a strong inductive bias for downstream tasks: interpolation and parameter estimation. That is, ContinuousCloth reduces interpolation error by 7.3 times over a data-only neural baseline in simulation and 2.4 times on real-world cloth sequences, and achieves 2.6 times lower parameter error than a differentiable-solver baseline under high-noise conditions, demonstrating consistent robustness under realistic measurement imperfections.

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