Liquid Random Feature Methods for Time-Dependent Partial Differential Equations

Abstract: 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.