Physics-Informed Neural Networks for Excited Liquid Sloshing with Beating Response in a Rectangular Tank

This paper applies physics-informed neural networks (PINNs) to laterally excited liquid sloshing in a two-dimensional rectangular tank, where near-resonant forcing (ωe/ω1 = 0.9) produces a multi frequency beating response with a period of approximately 10T1. Linearised potential flow theory governs the problem; the network learns the velocity potential φ(x,z,t) while the free-surface elevation η is injected analytically. Two training obstacles specific to forced sloshing are analysed. First, a zero solution trap arises because the trivial solution φ̂=0 satisfies all equations except the free-surface conditions, whose residuals are roughly 10−4 times smaller than the Laplace residual; characteristic scale normalisation combined with loss weighting (λD = λK = 100) breaks this trap. Second, spectral bias prevents standard MLPs from resolving the three co-existing frequencies (ω1, ωe, ∆ω); a Fourier time embedding that augments the input from 3 to 9 dimensions overcomes this limitation. Two additional techniques further reduce errors: a hard wall boundary condition enforced exactly via a cos(πx/B) spatial embedding, which eliminates wall collocation points; and a gradient-enhanced Laplace regulariser (∥∇(∇2φ̂)∥2) that constrains velocity smoothness through third-order automatic differentiation. An ablation study shows that these four techniques progressively reduce the horizontal velocity error from εu = 12.46% to 0.84%. Results are validated against a viscous finite-difference benchmark. Over one beating cycle the errors are εη = 0.15%, εu = 0.84%, and εw = 1.65%. Afrequency parameter study across ωe/ω1 = 0.5–1.1 gives εη < 0.25% and εu < 2.3% for all near-resonance cases. For long-time simulation, a time-domain decomposition strategy with transfer learning partitions the domain into one-beat windows; extending to five beating cycles (50T1) yields εu = 3.43% and εη = 0.30% with no monotonic error accumulation across windows. The methodology is then extended to a three-dimensional rectangular tank (B × W × H) with bi-directional lateral excitation. The 3-D formulation introduces the y-dimension into the Laplace equation (∇2φ = φxx + φyy + φzz = 0), adds transverse wall boundary conditions (∂φ/∂y = 0) enforced exactly via a cos(πy/W) embedding, and extends the Fourier time embedding from 9 to 16 dimensions to accommodate six physical frequencies. The bi-directional excitation excites both (m,0) and (0,n) modal families, producing a genuinely three dimensional beating response. Results demonstrate that the proposed techniques transfer effectively to 3-D, with errors εη = 0.24%, εu = 1.31%, εv = 1.78%, and εw = 2.32% over one beating cycle (2,499 s training time).