Antoine Caradot

Fayad Ali Banna, Antoine Caradot, Eduardo Brandao et al.

Identifying from observation data the governing differential equations of a physical dynamics is a key challenge in machine learning. Although approaches based on SINDy have shown great promise in this area, they still fail to address a whole class of real world problems where the data is sparsely sampled in time. In this article, we introduce Unrolled-SINDy, a simple methodology that leverages an unrolling scheme to improve the stability of explicit methods for PDE discovery. By decorrelating the numerical time step size from the sampling rate of the available data, our approach enables the recovery of equation parameters that would not be the minimizers of the original SINDy optimization problem due to large local truncation errors. Our method can be exploited either through an iterative closed-form approach or by a gradient descent scheme. Experiments show the versatility of our method. On both traditional SINDy and state-of-the-art noise-robust iNeuralSINDy, with different numerical schemes (Euler, RK4), our proposed unrolling scheme allows to tackle problems not accessible to non-unrolled methods.

Antoine Caradot, Rémi Emonet, Amaury Habrard et al.

Physics-informed Neural Networks (PINNs) have emerged as an efficient way to learn surrogate neural solvers of PDEs by embedding the physical model in the loss function and minimizing its residuals using automatic differentiation at so-called collocation points. Originally uniformly sampled, the choice of the latter has been the subject of recent advances leading to adaptive sampling refinements. In this paper, we propose a new quadrature method for approximating definite integrals based on the hessian of the considered function, and that we leverage to guide the selection of the collocation points during the training process of PINNs.

Antoine Caradot, Rémi Emonet, Amaury Habrard et al.

Despite considerable scientific advances in numerical simulation, efficiently solving PDEs remains a complex and often expensive problem. Physics-informed Neural Networks (PINN) have emerged as an efficient way to learn surrogate solvers by embedding the PDE in the loss function and minimizing its residuals using automatic differentiation at so-called collocation points. Originally uniformly sampled, the choice of the latter has been the subject of recent advances leading to adaptive sampling refinements for PINNs. In this paper, leveraging a new quadrature method for approximating definite integrals, we introduce a provably accurate sampling method for collocation points based on the Hessian of the PDE residuals. Comparative experiments conducted on a set of 1D and 2D PDEs demonstrate the benefits of our method.