Salah Eddine Choutri

Julian Barreiro-Gomez, Salah Eddine Choutri, Boualem Djehiche

In this paper, we present an approach to neural network mean-field-type control and its stochastic stability analysis by means of adversarial inputs (aka adversarial attacks). This is a class of data-driven mean-field-type control where the distribution of the variables such as the system states and control inputs are incorporated into the problem. Besides, we present a methodology to validate the feasibility of the approximations of the solutions via neural networks and evaluate their stability. Moreover, we enhance the stability by enlarging the training set with adversarial inputs to obtain a more robust neural network. Finally, a worked-out example based on the linear-quadratic mean-field type control problem (LQ-MTC) is presented to illustrate our methodology.

Prajwal Chauhan, Salah Eddine Choutri, Mohamed Ghattassi et al.

This paper investigates the limitations of neural operators in learning solutions for a Hughes model, a first-order hyperbolic conservation law system for crowd dynamics. The model couples a Fokker-Planck equation representing pedestrian density with a Hamilton-Jacobi-type (eikonal) equation. This Hughes model belongs to the class of nonlinear hyperbolic systems that often exhibit complex solution structures, including shocks and discontinuities. In this study, we assess the performance of three state-of-the-art neural operators (Fourier Neural Operator, Wavelet Neural Operator, and Multiwavelet Neural Operator) in various challenging scenarios. Specifically, we consider (1) discontinuous and Gaussian initial conditions and (2) diverse boundary conditions, while also examining the impact of different numerical schemes. Our results show that these neural operators perform well in easy scenarios with fewer discontinuities in the initial condition, yet they struggle in complex scenarios with multiple initial discontinuities and dynamic boundary conditions, even when trained specifically on such complex samples. The predicted solutions often appear smoother, resulting in a reduction in total variation and a loss of important physical features. This smoothing behavior is similar to issues discussed by Daganzo (1995), where models that introduce artificial diffusion were shown to miss essential features such as shock waves in hyperbolic systems. These results suggest that current neural operator architectures may introduce unintended regularization effects that limit their ability to capture transport dynamics governed by discontinuities. They also raise concerns about generalizing these methods to traffic applications where shock preservation is essential.

Salah Eddine Choutri, Prajwal Chauhan, Othmane Mazhar et al.

The Monte Carlo-type Neural Operator (MCNO) introduces a lightweight architecture for learning solution operators for parametric PDEs by directly approximating the kernel integral using a Monte Carlo approach. Unlike Fourier Neural Operators, MCNO makes no spectral or translation-invariance assumptions. The kernel is represented as a learnable tensor over a fixed set of randomly sampled points. This design enables generalization across multiple grid resolutions without relying on fixed global basis functions or repeated sampling during training. Experiments on standard 1D PDE benchmarks show that MCNO achieves competitive accuracy with low computational cost, providing a simple and practical alternative to spectral and graph-based neural operators.

Salah Eddine Choutri, Prajwal Chauhan, Othmane Mazhar et al.

The Monte Carlo-type Neural Operator (MCNO) introduces a framework for learning solution operators of one-dimensional partial differential equations (PDEs) by directly learning the kernel function and approximating the associated integral operator using a Monte Carlo-type approach. Unlike Fourier Neural Operators (FNOs), which rely on spectral representations and assume translation-invariant kernels, MCNO makes no such assumptions. The kernel is represented as a learnable tensor over sampled input-output pairs, and sampling is performed once, uniformly at random from a discretized grid. This design enables generalization across multiple grid resolutions without relying on fixed global basis functions or repeated sampling during training, while an interpolation step maps between arbitrary input and output grids to further enhance flexibility. Experiments on standard 1D PDE benchmarks show that MCNO achieves competitive accuracy with efficient computational cost. We also provide a theoretical analysis proving that the Monte Carlo estimator yields a bounded bias and variance under mild regularity assumptions. This result holds in any spatial dimension, suggesting that MCNO may extend naturally beyond one-dimensional problems. More broadly, this work explores how Monte Carlo-type integration can be incorporated into neural operator frameworks for continuous-domain PDEs, providing a theoretically supported alternative to spectral methods (such as FNO) and to graph-based Monte Carlo approaches (such as the Graph Kernel Neural Operator, GNO).