# Data driven approximation of parametrized PDEs by Reduced Basis and Neural Networks

arXiv: https://arxiv.org/abs/1904.01514 · HTML: https://mlsanity.com/paper/1904.01514/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Apr 2019.

## Built on by (most cited first, 22 of 22)

- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [Short-term traffic prediction using physics-aware neural networks](https://mlsanity.com/paper/2109.10253/) (2021)
- [JAX-DIPS: Neural bootstrapping of finite discretization methods and application to elliptic problems with discontinuities](https://mlsanity.com/paper/2210.14312/) (2022)
- [Learning smooth functions in high dimensions: from sparse polynomials to deep neural networks](https://mlsanity.com/paper/2404.03761/) (2024)
- [Variationally correct operator learning: Reduced basis neural operator with a posteriori error estimation](https://mlsanity.com/paper/2512.21319/) (2025)
- [CAS4DL: Christoffel Adaptive Sampling for function approximation via Deep Learning](https://mlsanity.com/paper/2208.12190/) (2022)
- [Solving parametric partial differential equations with deep rectified quadratic unit neural networks](https://mlsanity.com/paper/2203.06973/) (2022)
- [Data Assimilation Predictive GAN (DA-PredGAN): applied to determine the spread of COVID-19](https://mlsanity.com/paper/2105.07729/) (2021)
- [RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equations](https://mlsanity.com/paper/2509.01234/) (2025)
- [AI-enhanced iterative solvers for accelerating the solution of large scale parametrized systems](https://mlsanity.com/paper/2207.02543/) (2022)
- [Adaptive Multilevel Neural Networks for Parametric PDEs with Error Estimation](https://mlsanity.com/paper/2403.12650/) (2024)
- [DPG loss functions for learning parameter-to-solution maps by neural networks](https://mlsanity.com/paper/2506.18773/) (2025)
- [Data-Driven Constitutive Relation Reveals Scaling Law for Hydrodynamic Transport Coefficients](https://mlsanity.com/paper/2108.00413/) (2021)
- [Learning To Estimate Regions Of Attraction Of Autonomous Dynamical Systems Using Physics-Informed Neural Networks](https://mlsanity.com/paper/2111.09930/) (2021)
- [Neuro-symbolic partial differential equation solver](https://mlsanity.com/paper/2210.14907/) (2022)
- [Conditional variational autoencoder with Gaussian process regression recognition for parametric models](https://mlsanity.com/paper/2305.09625/) (2023)
- [Multilevel CNNs for Parametric PDEs based on Adaptive Finite Elements](https://mlsanity.com/paper/2408.10838/) (2024)
- [Reduced-Basis Deep Operator Learning for Parametric PDEs with Independently Varying Boundary and Source Data](https://mlsanity.com/paper/2511.18260/) (2025)
- [Addressing Discontinuous Root-Finding for Subsequent Differentiability in Machine Learning, Inverse Problems, and Control](https://mlsanity.com/paper/2306.12413/) (2023)
- [An Efficient Deep Learning Approach for Approximating Parameter-to-Solution Maps of PDEs](https://mlsanity.com/paper/2404.06834/) (2024)
- [Global error estimators for parametric monotone nonlinearities and neural approximations](https://mlsanity.com/paper/2610.05767/) (2026)

## Abstract (arXiv)

We are interested in the approximation of partial differential equations with a data-driven approach based on the reduced basis method and machine learning. We suppose that the phenomenon of interest can be modeled by a parametrized partial differential equation, but that the value of the physical parameters is unknown or difficult to be directly measured. Our method allows to estimate fields of interest, for instance temperature of a sample of material or velocity of a fluid, given data at a handful of points in the domain. We propose to accomplish this task with a neural network embedding a reduced basis solver as exotic activation function in the last layer. The reduced basis solver accounts for the underlying physical phenomenonon and it is constructed from snapshots obtained from randomly selected values of the physical parameters during an expensive offline phase. The same full order solutions are then employed for the training of the neural network. As a matter of fact, the chosen architecture resembles an asymmetric autoencoder in which the decoder is the reduced basis solver and as such it does not contain trainable parameters. The resulting latent space of our autoencoder includes parameter-dependent quantities feeding the reduced basis solver, which -- depending on the considered partial differential equation -- are the values of the physical parameters themselves or the affine decomposition coefficients of the differential operators.