# Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations

arXiv: https://arxiv.org/abs/2607.19377 · HTML: https://mlsanity.com/paper/2607.19377/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 35 references that are in the map.
- First posted on arXiv 1 Jul 2026.

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

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [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)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [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)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks](https://mlsanity.com/paper/2007.07442/) (2020)
- [Adaptive Hard-Soft Physics-Informed Neural Networks for Robust Boundary-Constrained PDE Solving](https://mlsanity.com/paper/2606.23359/) (2026)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) provide a mesh-free framework for solving partial differential equations, but their training is often affected by loss imbalance, optimization stiffness, and difficulty in capturing localized or multi-mode solution structures. Hard-soft PINNs (HSPINN) alleviate part of this difficulty by embedding Dirichlet or periodic constraints directly into the trial space, but the resulting fixed admissible representation can still be poorly conditioned for sharp or heterogeneous residual fields. This paper proposes a reliability-aware hard-soft PINN (RA-HSPINN) that preserves exact embedded constraints while introducing a bounded learnable reliability field to modulate the interior representation. The method combines this reliability-aware ansatz with inverse-EMA global loss balancing and lightweight regularization, while retaining the standard mean-square residual form. The reliability field is a numerical modulation variable, not a physical parameter or calibrated probability. RA-HSPINN is evaluated on nonlinear Burgers equations, periodic convection, a mixed-boundary Poisson problem, and a mixed first-order Poisson system. Compared with HSPINN, it reduces the relative error by 98.65% for sharp-gradient Burgers, 72.42% for Burgers data with noisy and incompatible initial conditions, 61.18% for smooth periodic convection, 60.02% for localized periodic convection, 29.36% for mixed-boundary Poisson, and 82.17% for a multi-mode mixed first-order Poisson system. The results show that reliability-aware modulation is most beneficial when hard-soft trial spaces are admissible but difficult to optimize, especially in localized, unreliable-data, and multi-mode PDE regimes.