# Feature tracking in physics-informed neural networks via joint optimization of nonlinear deformation manifolds: application to shocks

arXiv: https://arxiv.org/abs/2610.02230 · HTML: https://mlsanity.com/paper/2610.02230/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 38 references that are in the map.
- First posted on arXiv 24 Sep 2026.

## Builds on (most cited first, 15 of 15)

- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [Efficient training of physics-informed neural networks via importance sampling](https://mlsanity.com/paper/2104.12325/) (2021)
- [PirateNets: Physics-informed Deep Learning with Residual Adaptive Networks](https://mlsanity.com/paper/2402.00326/) (2024)
- [PINNACLE: PINN Adaptive ColLocation and Experimental points selection](https://mlsanity.com/paper/2404.07662/) (2024)
- [Physics informed WNO](https://mlsanity.com/paper/2302.05925/) (2023)
- [Physics-Informed Neural Networks with Adaptive Localized Artificial Viscosity](https://mlsanity.com/paper/2203.08802/) (2022)
- [PACMANN: Point Adaptive Collocation Method for Artificial Neural Networks](https://mlsanity.com/paper/2411.19632/) (2024)
- [A robust, high-order implicit shock tracking method for simulation of complex, high-speed flows](https://mlsanity.com/paper/2105.00139/) (2021)
- [An rp-adaptive method for accurate resolution of shock-dominated viscous flow based on implicit shock tracking](https://mlsanity.com/paper/2504.15177/) (2025)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) often converge to inaccurate solutions for conservation laws with shocks, because uniformly distributed collocation points undersample localized features and let the residual be dominated by regions that are already well resolved. We propose a feature-tracking PINN (FT-PINN) in which the solution network is defined on a fixed reference domain and composed with a diffeomorphic deformation map from a parameterized nonlinear manifold. The deformation and solution-network parameters are trained jointly by minimizing the pulled-back conservation-law residual. This lets collocation points concentrate along features of essentially arbitrary geometry, including curved, oblique, and merging shocks, without prior knowledge of their locations. The framework is agnostic to the choice of parameterization. Boundary preservation is enforced exactly through a tangential projection of the displacement, and folding is discouraged by a one-sided penalty on the Jacobian determinant. On four test problems (space-time viscous Burgers with merging shocks, a decelerating Burgers shock, the space-time Euler shock tube, and steady 2D Euler regular shock reflection), FT-PINN resolves shocks at their correct locations with a limited collocation budget. A vanilla PINN with the same architecture, budget, and training either misplaces the shocks or fails to form them.