arXiv subjects

Anas Jnini

Publications and source records attributed to Anas Jnini.

6 recordsLinked to original sources

arXiv2026

Curvature-Aware Optimization for High-Accuracy Physics-Informed Neural Networks

Efficient and robust optimization is essential for neural networks, enabling scientific machine learning models to converge rapidly to very high accuracy -- faithfully capturing complex physical behavior governed by differential equations. In this work, we present advanced optimization strategies to accelerate the convergence of physics-informed neural networks (PINNs) for challenging partial (PDEs) and ordinary differential equations (ODEs). Specifically, we provide efficient implementations of the Natural Gradient (NG) optimizer, Self-Scaling BFGS and Broyden optimizers, and demonstrate their performance on problems including the Helmholtz equation, Stokes flow, inviscid Burgers equation, Euler equations for high-speed flows, and stiff ODEs arising in pharmacokinetics and pharmacodynamics. Beyond optimizer development, we also propose new PINN-based methods for solving the inviscid Burgers and Euler equations, and compare the resulting solutions against high-order numerical methods to provide a rigorous and fair assessment. Finally, we address the challenge of scaling these quasi-Newton optimizers for batched training, enabling efficient and scalable solutions for large data-driven problems.

cs.LG↗

arXiv2025

Dual Natural Gradient Descent for Scalable Training of Physics-Informed Neural Networks

Natural-gradient methods markedly accelerate the training of Physics-Informed Neural Networks (PINNs), yet their Gauss--Newton update must be solved in the parameter space, incurring a prohibitive $O(n^3)$ time complexity, where $n$ is the number of network trainable weights. We show that exactly the same step can instead be formulated in a generally smaller residual space of size $m = \sum_{\gamma} N_{\gamma} d_{\gamma}$, where each residual class $\gamma$ (e.g. PDE interior, boundary, initial data) contributes $N_{\gamma}$ collocation points of output dimension $d_{\gamma}$. Building on this insight, we introduce \textit{Dual Natural Gradient Descent} (D-NGD). D-NGD computes the Gauss--Newton step in residual space, augments it with a geodesic-acceleration correction at negligible extra cost, and provides both a dense direct solver for modest $m$ and a Nystrom-preconditioned conjugate-gradient solver for larger $m$. Experimentally, D-NGD scales second-order PINN optimization to networks with up to 12.8 million parameters, delivers one- to three-order-of-magnitude lower final error $L^2$ than first-order methods (Adam, SGD) and quasi-Newton methods, and -- crucially -- enables natural-gradient training of PINNs at this scale on a single GPU.

cs.LG↗

arXiv2025

Physics-constrained DeepONet for Surrogate CFD models: a curved backward-facing step case

The Physics-Constrained DeepONet (PC-DeepONet), an architecture that incorporates fundamental physics knowledge into the data-driven DeepONet model, is presented in this study. This methodology is exemplified through surrogate modeling of fluid dynamics over a curved backward-facing step, a benchmark problem in computational fluid dynamics. The model was trained on computational fluid dynamics data generated for a range of parameterized geometries. The PC-DeepONet was able to learn the mapping from the parameters describing the geometry to the velocity and pressure fields. While the DeepONet is solely data-driven, the PC-DeepONet imposes the divergence constraint from the continuity equation onto the network. The PC-DeepONet demonstrates higher accuracy than the data-driven baseline, especially when trained on sparse data. Both models attain convergence with a small dataset of 50 samples and require only 50 iterations for convergence, highlighting the efficiency of neural operators in learning the dynamics governed by partial differential equations.

physics.flu-dyn↗

arXiv2025

Riemann Tensor Neural Networks: Learning Conservative Systems with Physics-Constrained Networks

Divergence-free symmetric tensors (DFSTs) are fundamental in continuum mechanics, encoding conservation laws such as mass and momentum conservation. We introduce Riemann Tensor Neural Networks (RTNNs), a novel neural architecture that inherently satisfies the DFST condition to machine precision, providing a strong inductive bias for enforcing these conservation laws. We prove that RTNNs can approximate any sufficiently smooth DFST with arbitrary precision and demonstrate their effectiveness as surrogates for conservative PDEs, achieving improved accuracy across benchmarks. This work is the first to use DFSTs as an inductive bias in neural PDE surrogates and to explicitly enforce the conservation of both mass and momentum within a physics-constrained neural architecture.

cs.LG↗

arXiv2024

cuVegas: Accelerate Multidimensional Monte Carlo Integration through a Parallelized CUDA-based Implementation of the VEGAS Enhanced Algorithm

This paper introduces cuVegas, a CUDA-based implementation of the Vegas Enhanced Algorithm (VEGAS+), optimized for multi-dimensional integration in GPU environments. The VEGAS+ algorithm is an advanced form of Monte Carlo integration, recognized for its adaptability and effectiveness in handling complex, high-dimensional integrands. It employs a combination of variance reduction techniques, namely adaptive importance sampling and a variant of adaptive stratified sampling, that make it particularly adept at managing integrands with multiple peaks or those aligned with the diagonals of the integration volume. Being a Monte Carlo integration method, the task is well suited for parallelization and for GPU execution. Our implementation, cuVegas, aims to harness the inherent parallelism of GPUs, addressing the challenge of workload distribution that often hampers efficiency in standard implementations. We present a comprehensive analysis comparing cuVegas with existing CPU and GPU implementations, demonstrating significant performance improvements, from two to three orders of magnitude on CPUs, and from a factor of two on GPUs over the best existing implementation. We also demonstrate the speedup for integrands for which VEGAS+ was designed, with multiple peaks or other significant structures aligned with diagonals of the integration volume.

cs.DC↗

arXiv2024

Gauss-Newton Natural Gradient Descent for Physics-Informed Computational Fluid Dynamics

We propose Gauss-Newton's method in function space for the solution of the Navier-Stokes equations in the physics-informed neural network (PINN) framework. Upon discretization, this yields a natural gradient method that provably mimics the function space dynamics. Our computational results demonstrate close to single-precision accuracy measured in relative $L^2$ norm on a number of benchmark problems. To the best of our knowledge, this constitutes the first contribution in the PINN literature that solves the Navier-Stokes equations to this degree of accuracy. Finally, we show that given a suitable integral discretization, the proposed optimization algorithm agrees with Gauss-Newton's method in parameter space. This allows a matrix-free formulation enabling efficient scalability to large network sizes.

math.OC↗