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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.23409 |
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| _version_ | 1866915644130197504 |
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| author | Abbas, Naseem Colao, Vittorio Macri, Davide Spataro, William |
| author_facet | Abbas, Naseem Colao, Vittorio Macri, Davide Spataro, William |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_23409 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Multi-Phase Dual-PINN Framework: Soft Boundary-Interior Specialization via Distance-Weighted Priors Abbas, Naseem Colao, Vittorio Macri, Davide Spataro, William Numerical Analysis 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. |
| title | A Multi-Phase Dual-PINN Framework: Soft Boundary-Interior Specialization via Distance-Weighted Priors |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2511.23409 |