Physics-Informed Neural Networks for Radial Consolidation of Combined Electroosmotic, Vacuum and Surcharge Preloading Considering Smear Effects

Physics-informed neural networks (PINNs) for coupled multiphysics systems suffer systematic accuracy degradation as inter-equation coupling strengthens. We provide a theoretical explanation through neural tangent kernel (NTK) analysis: for linearly coupled systems, the standard NTK's spectral radius grows as $Ω(γ^2)$ with coupling strength $γ$, shrinking the stable learning rate, while block-diagonal Gauss--Newton (GN) preconditioning yields a preconditioned NTK $K_P = JH^{+}J^\top$ whose spectral radius is bounded by $S$ (number of networks), independent of $γ$. Adam's diagonal preconditioning destroys this projector structure -- inflating $λ_{\max}$ far above $S$ for any coupling type -- and its residual-dynamics kernel grows as $Θ(γ)$, placing its stable learning rate strictly between gradient descent and GN. For one-way coupling the limitation is class-wide: no diagonal preconditioner, fixed or adaptive, halves the driving residual in fewer than $Ω(γ)$ iterations ($Ω(γ^2)$ if fixed), whereas block-diagonal GN requires $O(1)$. We verify $Ω(γ^2)$ growth across linearly coupled benchmarks and confirm $λ_{\max}(K_P) = S$ in all three 1D systems, including nonlinearly coupled NP+P. Combining the Kronecker-preconditioned optimizer SOAP with inverse-gradient-norm loss balancing (SOAP+GradNorm) yields coupling-robust accuracy: across 222 experiments spanning three 1D systems and a 2D electroosmotic flow benchmark, SOAP+GradNorm maintains final-epoch $L_2$ accuracy across coupling strengths, with $\leq 2.3\times$ degradation in nonlinear NP+P while Adam+GradNorm fails ($L_2 > 0.1$). SOAP+GradNorm further scales to a 2D, 6-PDE electroosmotic flow at EDL-resolved conditions down to $\varepsilon = 0.01$ -- a regime all prior PINN electrokinetics studies have avoided -- where Adam+GradNorm fails entirely ($L_2 > 0.3$).

Physics-Informed Neural Networks (PINNs) are a machine learning method for solving forward and inverse Partial Differential Equations (PDEs). When applied to PDEs with Dirac delta functions in the forcing terms, boundary conditions, or initial conditions, PINNs require approximating them with smooth surrogate functions, a practice that can introduce significant modeling errors. In this work, we exploit the interpretation of PINNs as Residual Least Squares (RLS) methods and show that this perspective enables direct treatment of Dirac delta terms by integrating the weak-form equation. Among RLS formulations other than PINN, we focus on the Radial Basis Function (RBF) expansion (also known as a single-layer RBF Network). We show that while integrating out the Dirac delta in PINNs causes residuals to fail to converge to zero, RBF-RLS consistently provides good forward and inverse solutions to transport problems. We explain this finding using the Neural Tangent Kernel (NTK) theory. We test both approaches on linear PDEs that represent groundwater flow and transport in porous media and rivers. We solve inverse problems to fit synthetic data, noisy synthetic data, and real-world measurements.

Physics-informed neural networks (PINNs) have become a powerful framework for solving PDEs by embedding physical laws into differentiable objectives. Despite their advances, training PINNs remains fragile: recent conflict-averse optimization schemes alleviate gradient interference between residual and boundary losses, but we show that their effectiveness deteriorates as model capacity increases. In this paper, we identify a capacity-induced failure mode, where overparameterized networks undergo functional modularity, self-partitioning into task-exclusive modules that suppress cross-objective interaction and hinder convergence toward Pareto-stationary points. To address this issue, we propose a novel framework, Modular-Sparsity Synchronization (ModSync), which integrates structural optimization into conflict-averse training by penalizing task-exclusive connections while preserving interaction-promoting pathways. Extensive experiments across diverse PDE benchmarks demonstrate that ModSync consistently prevents capacity-driven failures, sustains robust cross-objective coupling, and achieves state-of-the-art accuracy. Codes are available at \url{https://github.com/heejokong/ModSync}.