Yangshuai Wang

Xi'an Li, Jiale Linghu, Yangshuai Wang

Data assimilation for time-dependent partial differential equations (PDEs) requires Bayesian updates of an evolving field from streaming, sparse, and noisy observations, while keeping the filtering state finite dimensional. We introduce a random-feature Kalman filtering framework for linear PDE data assimilation. Once the random features are frozen and the linear PDE is Galerkin discretized, the coefficient vector satisfies a finite-dimensional linear-Gaussian state-space model, so the Kalman recursion gives the exact posterior for the chosen coefficient model. For non-orthogonal random-feature draws, we construct a mass-whitened effective-rank coordinate system that removes near-null mass directions and identifies the posterior dimension $r$. For the heat equation with implicit-Euler time stepping, we prove a high-probability posterior-contraction and PDE-consistency theorem in these mass-whitened coordinates. The mean-square $L^2$ reconstruction error separates into an effective-rank feature approximation term, a deterministic time-consistency term, and a Bayesian estimation term. In the high-information regime, the leading posterior contribution scales as $rσ^2/N_o$, where $σ^2$ is the observation-noise variance and $N_o$ is the number of observations per analysis time. Thus the analysis distinguishes the exact coefficient-space posterior from deterministic PDE approximation errors, and gives a checkable uncertainty-quantification guarantee for random-feature filtering of a representative parabolic PDE.

Jiale Linghu, Yangshuai Wang

A central challenge in mesh-free space--time approximation for time-dependent partial differential equations is to represent evolving temporal scales while keeping residual minimization computationally tractable. Random feature methods simplify this algebraic problem by freezing nonlinear trial functions and fitting only a linear readout, but standard static space--time activations provide no explicit relaxation-scale mechanism, making temporal-scale resolution a finite-dimensional bottleneck in stiff, dispersive, or multi-scale regimes. We introduce liquid random feature methods (L-RFM), which replace static temporal activations by closed-form liquid time-constant responses with sampled relaxation scales. The resulting frozen features form temporally structured local or global trial spaces with analytic space--time derivatives for residual least-squares assembly. A density theorem proves density of the deterministic trial spaces in the continuous space--time function class, and a temporal-rank calculation clarifies the role of sampled relaxation scales. Ablation and finite-feature tests identify the liquid temporal response as the primary source of the observed accuracy improvement. Across stiff reaction--diffusion, nonlinear transport, dispersive, complex-valued, and multidimensional benchmarks, L-RFM improves finite-feature accuracy in regimes where temporal-scale representation controls the approximation. By embedding relaxation scales directly into frozen trial functions, L-RFM provides a route to high-accuracy continuous space--time surrogates for evolutionary PDEs while preserving the simplicity of linear least-squares solvers.