# Physics-informed neural networks for multidisciplinary analysis and design

Nathan Ricard, Mickaël Binois, Régis Duvigneau

► To cite this version:

Nathan Ricard, Mickaël Binois, Régis Duvigneau. Physics-informed neural networks for multidisciplinary analysis and design. SAMO 2025: International Conference on Sensitivity Analysis of Model Output, Apr 2025, Grenoble, France. hal-05361355

HAL Id: hal-05361355

https://inria.hal.science/hal-05361355v1

Submitted on 12 Nov 2025

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Distributed under a Creative Commons Attribution 4.0 International License
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# Physics-Informed Neural Networks
## for multidisciplinary analysis and design

<figure><img src="image_3.png" /></figure>

Nathan Ricard

Université Côte d'Azur, Inria, CNRS, LJAD, nathan.ricard@inria.fr (supervisors: R. Duvigneau and M. Binois)

## 1. PDE problem

* Generally, a PDE problem is:

$$ egin{cases} 
\frac{\partial u}{\partial t} + \mathcal{N}[u] = 0, (t, x) \in [0, T] \times \Omega, \\ 
u(0, x) = g(x), x \in \Omega, \\ 
\mathcal{B}[u] = 0, (t, x) \in [0, T] \times \partial\Omega, 
\end{cases} $$

* Objective: approximate $u$ with a neural network $u_\vartheta$, where $\vartheta$ denotes all tunable parameters.

## 2. PINNs overview

<figure><img src="image_4.png" /></figure>

* The main interests of PINNs are the following:

- resolution of **parametric problems** (generalize predictions for design or uncertainty quantification);

- resolution of **inverse problems** (reconstruct unknown physical parameters);

- resolution of **mixed problems** using both physical PDE models and experimental data.

* But some issues remain:

- Choice of hyperparameters: network architecture, optimization algorithm, activation functions, etc.;

- Large computational time due to the resolution of a large scale optimization problem during the learning;

- Convergence failures or low convergence rate.

## 4. Study of convergence issues

<figure><img src="image_5.png" /><figcaption>Ill-conditioning of the Hessian matrix of the loss function</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Conflictual gradients of the loss function terms</figcaption></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

* Anisotropic problem yielding a **low efficiency of first-order descent methods** like Adam;

* The gradients of the loss function terms related to differential residuals and boundary conditions are almost opposite;

* Use of adaptive weights to improve the training with Adam or projected gradients not really efficient in general.

## 3. Heated cavity problem

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>(a) Reference - Temperature</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(b) Reference - Velocity Norm</figcaption></figure>

## 5. Research axis : Nash games

* Splitting of territories : $\vartheta = (\vartheta_c, \vartheta_p, \vartheta_b)$,

* Transform the classic problem : $\min_{\vartheta} \mathcal{L}_{PINN}$ to:

$$ egin{cases} 
\min_{\vartheta_c} \mathcal{L}_{PINN} \\ 
\min_{\vartheta_p} \mathcal{L}_{PDE} \\ 
\min_{\vartheta_b} \mathcal{L}_{BC} 
\end{cases} $$

* $\vartheta_c = \{\vartheta_i\}$ such as :

$$ \frac{\partial \mathcal{L}_{PDE}}{\partial \vartheta_i} \cdot \frac{\partial \mathcal{L}_{BC}}{\partial \vartheta_i} > 0, \quad (1) $$

* $\vartheta_p = \{\vartheta_i\}$ such as (1) not satisfied and :

$$ \frac{\left| \frac{\partial \mathcal{L}_{PDE}}{\partial \vartheta_i} \right|}{\| \nabla_{\vartheta} \mathcal{L}_{PDE} \|} > \frac{\left| \frac{\partial \mathcal{L}_{BC}}{\partial \vartheta_i} \right|}{\| \nabla_{\vartheta} \mathcal{L}_{BC} \|}, \quad (2) $$

* $\vartheta_b = \{\vartheta_i\}$ such as (1) and (2) not satisfied,

* Illustration for Laplace 1D problem:

$$ \frac{d^2 u}{dx^2} = -(4 \sin(2x) + \cos(x)), x \in [0, 5] $$

$$ u(0) = 1, u(5) = -1. $$

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

## References

[1] S. Krishnapriyan, A. Gholami, S. Zhe, R.M. Kirby, M.W. Mahoney, "Characterizing possible failure modes in physics-informed neural networks", 2021.

[2] M. Raissi, P. Perdikaris, G.E. Karniadakis, "Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations", *Journal of Computational Physics*, 378:686–707, February 2019.

[3] Y. Liu, X. Li, Z. Liu, "An Improved Physics-Informed Neural Network Based on a New Adaptive Gradient Descent Algorithm for Solving Partial Differential Equations of Nonlinear Systems", November 2022.