# Hybrid Adaptive Modeling in Process Monitoring: Leveraging Sequence Encoders and Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2505.14252 · HTML: https://mlsanity.com/paper/2505.14252/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 36 references that are in the map.
- First posted on arXiv 20 May 2025.

## Builds on (most cited first, 14 of 14)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [Forecast Evaluation for Data Scientists: Common Pitfalls and Best Practices](https://mlsanity.com/paper/2203.10716/) (2022)
- [Any equation is a forest: Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE)](https://mlsanity.com/paper/2106.11927/) (2021)
- [Fusing Physics-based and Deep Learning Models for Prognostics](https://mlsanity.com/paper/2003.00732/) (2020)
- [A Unified Framework for Sparse Relaxed Regularized Regression: SR3](https://mlsanity.com/paper/1807.05411/) (2018)
- [Multifidelity Modeling for Physics-Informed Neural Networks (PINNs)](https://mlsanity.com/paper/2106.13361/) (2021)
- [Adaptive parameters identification for nonlinear dynamics using deep permutation invariant networks](https://mlsanity.com/paper/2501.11350/) (2025)

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

- [Physics-Informed Neural Networks and Sequence Encoder: Application to heating and early cooling of thermo-stamping process](https://mlsanity.com/paper/2603.26245/) (2026)

## Abstract (arXiv)

In this work, we explore the integration of Sequence Encoding for Online Parameter Identification with Physics-Informed Neural Networks to create a model that, once trained, can be utilized for real time applications with variable parameters, boundary conditions, and initial conditions. Recently, the combination of PINNs with Sparse Regression has emerged as a method for performing dynamical system identification through supervised learning and sparse regression optimization, while also solving the dynamics using PINNs. However, this approach can be limited by variations in parameters or boundary and initial conditions, requiring retraining of the model whenever changes occur. In this work, we introduce an architecture that employs Deep Sets or Sequence Encoders to encode dynamic parameters, boundary conditions, and initial conditions, using these encoded features as inputs for the PINN, enabling the model to adapt to changes in parameters, BCs, and ICs. We apply this approach to three different problems. First, we analyze the Rossler ODE system, demonstrating the robustness of the model with respect to noise and its ability to generalize. Next, we explore the model's capability in a 2D Navier-Stokes PDE problem involving flow past a cylinder with a parametric sinusoidal inlet velocity function, showing that the model can encode pressure data from a few points to identify the inlet velocity profile and utilize physics to compute velocity and pressure throughout the domain. Finally, we address a 1D heat monitoring problem using real data from the heating of glass fiber and thermoplastic composite plates.