# Adaptive Hard-Soft Physics-Informed Neural Networks for Robust Boundary-Constrained PDE Solving

arXiv: https://arxiv.org/abs/2606.23359 · HTML: https://mlsanity.com/paper/2606.23359/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 32 references that are in the map.
- First posted on arXiv 22 Jun 2026.

## Builds on (most cited first, 11 of 11)

- [GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks](https://mlsanity.com/paper/1711.02257/) (2017)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Learning to Accelerate Partial Differential Equations via Latent Global Evolution](https://mlsanity.com/paper/2206.07681/) (2022)
- [Parameterized Physics-informed Neural Networks for Parameterized PDEs](https://mlsanity.com/paper/2408.09446/) (2024)
- [A Survey on Activation Functions and their relation with Xavier and He Normal Initialization](https://mlsanity.com/paper/2004.06632/) (2020)
- [Optimizing the optimizer for data driven deep neural networks and physics informed neural networks](https://mlsanity.com/paper/2205.07430/) (2022)

## Built on by (most cited first, 1 of 1)

- [Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations](https://mlsanity.com/paper/2607.19377/) (2026)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) provide an effective way to solve partial differential equations (PDEs) by embedding physical principles into the learning process. However, the conventional PINN formulation, in which all constraints are imposed as soft penalty terms within a composite loss, often exhibits slow convergence, sensitivity to loss weight scaling, and inaccurate boundary enforcement due to poor conditioning of the optimization landscape. To address these limitations, this study proposes a unified hard--soft physics--informed neural network (HSPINN) with adaptive loss weighting. In this framework, Dirichlet and periodic boundary conditions are enforced exactly by construction through analytical or polynomial lifting, masking functions, and periodic feature mappings, while the governing PDE residuals, Neumann fluxes, and initial conditions are treated as soft constraints. An inverse-share softmax strategy dynamically balances the relative importance of individual loss components during training, eliminating manual penalty tuning and improving gradient stability. This formulation ensures boundary admissibility throughout optimization and enhances convergence efficiency and numerical robustness. Applications to representative elliptic (Poisson), parabolic (Burgers), and hyperbolic (convection with periodic boundaries) problems demonstrate that HSPINN consistently achieves faster convergence, higher accuracy, and greater stability than conventional PINNs, establishing a general and scalable foundation for physics-constrained deep learning across science and technology.