# Active Training of Physics-Informed Neural Networks to Aggregate and Interpolate Parametric Solutions to the Navier-Stokes Equations

arXiv: https://arxiv.org/abs/2005.05092 · HTML: https://mlsanity.com/paper/2005.05092/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 May 2020.

## Built on by (most cited first, 24 of 24)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
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- [Active Learning for Neural PDE Solvers](https://mlsanity.com/paper/2408.01536/) (2024)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations](https://mlsanity.com/paper/2210.07880/) (2022)
- [Active Learning with Selective Time-Step Acquisition for PDEs](https://mlsanity.com/paper/2511.18107/) (2025)
- [An AI-based Domain-Decomposition Non-Intrusive Reduced-Order Model for Extended Domains applied to Multiphase Flow in Pipes](https://mlsanity.com/paper/2202.06170/) (2022)
- [Temporal Consistency Loss for Physics-Informed Neural Networks](https://mlsanity.com/paper/2301.13262/) (2023)
- [Physics-Coupled Spatio-Temporal Active Learning for Dynamical Systems](https://mlsanity.com/paper/2108.05385/) (2021)
- [Investigation of Physics-Informed Deep Learning for the Prediction of Parametric, Three-Dimensional Flow Based on Boundary Data](https://mlsanity.com/paper/2203.09204/) (2022)
- [Physics-Informed Machine Learning of Argon Gas-Driven Melt Pool Dynamics](https://mlsanity.com/paper/2307.12304/) (2023)
- [Investigating the Ability of PINNs To Solve Burgers' PDE Near Finite-Time BlowUp](https://mlsanity.com/paper/2310.05169/) (2023)
- [Active Learning for Deep Learning-Based Hemodynamic Parameter Estimation](https://mlsanity.com/paper/2503.03453/) (2025)
- [Noisy PDE Training Requires Bigger PINNs](https://mlsanity.com/paper/2507.06967/) (2025)
- [Design of Turing Systems with Physics-Informed Neural Networks](https://mlsanity.com/paper/2211.13464/) (2022)
- [Investigation of Compressor Cascade Flow Using Physics- Informed Neural Networks with Adaptive Learning Strategy](https://mlsanity.com/paper/2308.04501/) (2023)
- [Ensemble models outperform single model uncertainties and predictions for operator-learning of hypersonic flows](https://mlsanity.com/paper/2311.00060/) (2023)
- [Improving PINNs By Algebraic Inclusion of Boundary and Initial Conditions](https://mlsanity.com/paper/2407.20741/) (2024)
- [An efficient hp-Variational PINNs framework for incompressible Navier-Stokes equations](https://mlsanity.com/paper/2409.04143/) (2024)
- [Learning Null Geodesics for Gravitational Lensing Rendering in General Relativity](https://mlsanity.com/paper/2507.15775/) (2025)
- [Intelligent Optimization of Multi-Parameter Micromixers Using a Scientific Machine Learning Framework](https://mlsanity.com/paper/2511.07702/) (2025)
- [Surrogate models for Rock-Fluid Interaction: A Grid-Size-Invariant Approach](https://mlsanity.com/paper/2602.22188/) (2026)
- [A Numerical Method for Coupling Parameterized Physics-Informed Neural Networks and FDM for Advanced Thermal-Hydraulic System Simulation](https://mlsanity.com/paper/2604.02663/) (2026)
- [Loss-Conditional PINNs for Parametric PDE Families](https://mlsanity.com/paper/2606.04420/) (2026)

## Abstract (arXiv)

The goal of this work is to train a neural network which approximates solutions to the Navier-Stokes equations across a region of parameter space, in which the parameters define physical properties such as domain shape and boundary conditions. The contributions of this work are threefold: 1) To demonstrate that neural networks can be efficient aggregators of whole families of parameteric solutions to physical problems, trained using data created with traditional, trusted numerical methods such as finite elements. Advantages include extremely fast evaluation of pressure and velocity at any point in physical and parameter space (asymptotically, ~3 μ s / query), and data compression (the network requires 99% less storage space compared to its own training data). 2) To demonstrate that the neural networks can accurately interpolate between finite element solutions in parameter space, allowing them to be instantly queried for pressure and velocity field solutions to problems for which traditional simulations have never been performed. 3) To introduce an active learning algorithm, so that during training, a finite element solver can automatically be queried to obtain additional training data in locations where the neural network's predictions are in most need of improvement, thus autonomously acquiring and efficiently distributing training data throughout parameter space. In addition to the obvious utility of Item 2, above, we demonstrate an application of the network in rapid parameter sweeping, very precisely predicting the degree of narrowing in a tube which would result in a 50% increase in end-to-end pressure difference at a given flow rate. This capability could have applications in both medical diagnosis of arterial disease, and in computer-aided design.