Physics-Informed Neural Networks for Excited Liquid Sloshing with Beating Response in Two- and Three-Dimensional Rectangular Tanks

This paper applies physics-informed neural networks (PINNs) to laterally excited liquid sloshing in a two-dimensional rectangular tank, where near-resonant forcing (${\omega }_e/{\omega }_1=0.9$) produces a multi-frequency beating response with a period of approximately $10T_1$. Linearized potential flow theory governs the problem; the network learns the velocity potential $\phi (x,z,t)$ while the free-surface elevation $\eta$ is injected analytically. Two training obstacles specific to forced sloshing are analyzed. First, a zero-solution trap arises because the trivial solution $\widehat{\phi }=0$ satisfies all equations except the free-surface conditions, whose residuals are roughly ${10}^{-4}$ times smaller than the Laplace residual; characteristic-scale normalization combined with loss weighting (${\lambda }_D={\lambda }_K=100$) breaks this trap. Second, spectral bias prevents standard MLPs from resolving the three co-existing frequencies (${\omega }_1$, ${\omega }_e$, $\Delta \omega$); 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 $cos(\pi x/B)$ spatial embedding, which eliminates wall collocation points; and a gradient-enhanced Laplace regularizer ($\parallel \nabla \left({\nabla }^2\widehat{\phi }\right){\parallel }^2$) that constrains velocity smoothness through third-order automatic differentiation. An ablation study shows that these four techniques progressively reduce the horizontal velocity error from ${\epsilon }_u=12.46\%$ to $0.84\%$. Results are validated against a viscous finite-difference benchmark. Over one beating cycle the errors are ${\epsilon }_{\eta }=0.15\%$, ${\epsilon }_u=0.84\%$, and ${\epsilon }_w=1.65\%$. A frequency parameter study across ${\omega }_e/{\omega }_1$ = 0.5–1.1 gives ${\epsilon }_{\eta }<0.25\%$ and ${\epsilon }_u<2.3\%$ 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 ($50T_1$) yields ${\epsilon }_u=3.43\%$ and ${\epsilon }_{\eta }=0.30\%$ with no monotonic error accumulation across windows. The methodology is then extended to a three-dimensional rectangular tank ($B\times W\times H$) with bi-directional lateral excitation. The 3-D formulation introduces the y-dimension into the Laplace equation (${\nabla }^2\phi ={\phi }_{xx}+{\phi }_{yy}+{\phi }_{zz}=0$), adds transverse wall boundary conditions ($\partial \phi /\partial y=0$) enforced exactly via a $cos(\pi y/W)$ 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. Experimental results verify that the proposed methods can be well generalized to three-dimensional scenarios. Within a single beating cycle, the relative errors reach ${\epsilon }_{\eta }=0.24\%$, ${\epsilon }_u=1.31\%$, ${\epsilon }_v=1.78\%$ and ${\epsilon }_w=2.32\%$, with a total training time of 2499 s. By applying time domain decomposition to carry out two-cycle three-dimensional simulations, the model can steadily maintain satisfactory prediction precision across segmented time intervals, achieving overall errors of ${\epsilon }_{\eta }=0.30\%$ and ${\epsilon }_u=1.32\%$.