# State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learning

arXiv: https://arxiv.org/abs/2202.06416 · HTML: https://mlsanity.com/paper/2202.06416/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 145 references that are in the map.
- First posted on arXiv 13 Feb 2022.

## Builds on (most cited first, 16 of 16)

- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [EikoNet: Solving the Eikonal equation with Deep Neural Networks](https://mlsanity.com/paper/2004.00361/) (2020)
- [nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applications](https://mlsanity.com/paper/2004.04276/) (2020)
- [On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs](https://mlsanity.com/paper/2004.01806/) (2020)
- [Multifidelity Modeling for Physics-Informed Neural Networks (PINNs)](https://mlsanity.com/paper/2106.13361/) (2021)
- [Conditional physics informed neural networks](https://mlsanity.com/paper/2104.02741/) (2021)
- [Polynomial-Chaos-based Kriging](https://mlsanity.com/paper/1502.03939/) (2015)
- [Physics-informed Neural Networks for Solving Nonlinear Diffusivity and Biot's equations](https://mlsanity.com/paper/2002.08235/) (2020)
- [A deep learning framework for solution and discovery in solid mechanics](https://mlsanity.com/paper/2003.02751/) (2020)
- [A transfer learning enhanced the physics-informed neural network model for vortex-induced vibration](https://mlsanity.com/paper/2112.14448/) (2021)

## Built on by (most cited first, 12 of 12)

- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs](https://mlsanity.com/paper/2306.08827/) (2023)
- [RANG: A Residual-based Adaptive Node Generation Method for Physics-Informed Neural Networks](https://mlsanity.com/paper/2205.01051/) (2022)
- [RBF-PINN: Non-Fourier Positional Embedding in Physics-Informed Neural Networks](https://mlsanity.com/paper/2402.08367/) (2024)
- [Physics-informed Neural Networks for Functional Differential Equations: Cylindrical Approximation and Its Convergence Guarantees](https://mlsanity.com/paper/2410.18153/) (2024)
- [Robust Regression with Highly Corrupted Data via Physics Informed Neural Networks](https://mlsanity.com/paper/2210.10646/) (2022)
- [How to Do Machine Learning with Small Data? -- A Review from an Industrial Perspective](https://mlsanity.com/paper/2311.07126/) (2023)
- [Feature Mapping in Physics-Informed Neural Networks (PINNs)](https://mlsanity.com/paper/2402.06955/) (2024)
- [Scientific Machine Learning Seismology](https://mlsanity.com/paper/2409.18397/) (2024)
- [Spectral-Bias and Kernel-Task Alignment in Physically Informed Neural Networks](https://mlsanity.com/paper/2307.06362/) (2023)
- [Explainable Multi-View Deep Networks Methodology for Experimental Physics](https://mlsanity.com/paper/2308.08206/) (2023)

## Abstract (arXiv)

This paper presents a comprehensive review of the design of experiments used in the surrogate models. In particular, this study demonstrates the necessity of the design of experiment schemes for the Physics-Informed Neural Network (PINN), which belongs to the supervised learning class. Many complex partial differential equations (PDEs) do not have any analytical solution; only numerical methods are used to solve the equations, which is computationally expensive. In recent decades, PINN has gained popularity as a replacement for numerical methods to reduce the computational budget. PINN uses physical information in the form of differential equations to enhance the performance of the neural networks. Though it works efficiently, the choice of the design of experiment scheme is important as the accuracy of the predicted responses using PINN depends on the training data. In this study, five different PDEs are used for numerical purposes, i.e., viscous Burger's equation, Shr"odinger equation, heat equation, Allen-Cahn equation, and Korteweg-de Vries equation. A comparative study is performed to establish the necessity of the selection of a DoE scheme. It is seen that the Hammersley sampling-based PINN performs better than other DoE sample strategies.