A Multi-Phase Dual-PINN Framework: Soft Boundary-Interior Specialization via Distance-Weighted Priors

TL;DR

This work introduces Dual-PINN, a two-network physics-informed framework that decomposes a PDE solution into interior and near-boundary components while sharing a single physics residual. Soft boundary-interior specialization via distance-weighted priors and cosine-annealed role weights guides early learning, and Dirichlet boundary conditions are enforced with an augmented Lagrangian method to reduce tuning effort. A two-phase curriculum—uniform interior sampling followed by boundary- or residual-focused sampling—enables efficient resolution of localized features without excessive computation. Across Laplace, Poisson, and 1D Fokker-Planck benchmarks, Dual-PINN yields substantial improvements over single-network PINNs, with ablations confirming the benefits of soft specialization, annealing, and the two-phase strategy. The approach is simple to implement, scalable, and broadly applicable to PDEs with sharp gradients or complex boundary data.

Abstract

Physics-informed neural networks (PINNs) often struggle with multi-scale PDEs featuring sharp gradients and nontrivial boundary conditions, as the physics residual and boundary enforcement compete during optimization. We present a dual-network framework that decomposes the solution as $u = u_{\text{D}} + u_{\text{B}}$, where $u_{\text{D}}$ (domain network) captures interior dynamics and $u_{\text{B}}$ (boundary network) handles near-boundary corrections. Both networks share a unified physics residual while being softly specialized via distance-weighted priors ($w_{\text{bd}} = \exp(-d/τ)$) that are cosine-annealed during training. Boundary conditions are enforced through an augmented Lagrangian method, eliminating manual penalty tuning. Training proceeds in two phases: Phase~1 uses uniform collocation to establish network roles and stabilize boundary satisfaction; Phase~2 employs focused sampling (e.g. ring sampling near $\partialΩ$) with annealed role weights to efficiently resolve localized features. We evaluate our model on four benchmarks, including the 1D Fokker-Planck equation, the Laplace equation, the Poisson equation, and the 1D wave equation. Across Laplace and Poisson benchmarks, our method reduces error by $36-90\%$, improves boundary satisfaction by $21-88\%$, and decreases MAE by $2.2-9.3\times$ relative to a single-network PINN. Ablations isolate contributions of (i)~soft boundary-interior specialization, (ii)~annealed role regularization, and (iii)~the two-phase curriculum. The method is simple to implement, adds minimal computational overhead, and broadly applies to PDEs with sharp solutions and complex boundary data.

Figures (8)

• Figure 1: Dual-PINN with two networks and soft boundary--interior roles for the 1D Fokker--Planck equation (configuration of Table \ref{['FP_2+2']}). Left: comparison between the exact solution (black dashed) and the combined prediction $u_{\mathrm D}+u_{\mathrm B}$ (red). Right: role decomposition. The "boundary" subnetwork $u_{\mathrm B}$ (green) remains active across the whole domain, while the "domain" subnetwork $u_{\mathrm D}$ (blue) produces small oscillatory corrections, showing that the desired specialization does not emerge in this joint ALM training.
• Figure 2: Dual-network residual PINN for the 1D Fokker--Planck equation. The black dashed line is the analytical solution, while the red line shows the combined prediction $u_{\mathrm B}+u_{\mathrm D}$. The green and blue curves indicate respectively the contributions of the baseline subnetwork $u_{\mathrm B}$ and of the residual subnetwork $u_{\mathrm D}$. The residual network primarily refines the internal structure, while the baseline controls the overall shape.
• Figure 3: Comparison along slice $y=0.8$ and statistical summary of evaluation metrics with seed $46$.
• Figure 4: Comparison along slice $y=0.8$ and statistical summary of evaluation metrics with seed $48$.
• Figure 5: Comparison along slice $y=0.8$ and statistical summary of evaluation metrics with seed $42$.