arXiv Machine Learning August 6, 2026 By Denis Gueyffier (ONERA -- Institut Polytechnique de Paris)

Guarantees by Construction for Learned Finite Volume Schemes on Steady Supersonic Flow

Read the original on arXiv Machine Learning →

arXiv:2607. 20171v3 Announce Type: replace-cross Abstract: A second order finite volume scheme rests on two local quantities: a gradient reconstructed in each cell, and a limiter which scales it down where the reconstruction would overshoot.

Summary generated by The Flow from the publisher's feed. The full article lives at arXiv Machine Learning.

arXiv Machine Learning

Jul 23

Hard Guarantees at a Measured Price: Entropy-Stable Learned Finite Volumes for Compressible Flow

arXiv:2607. 20171v1 Announce Type: cross Abstract: Learned solvers for compressible flow are usually compared to classical methods at equal mesh resolution rather than at equal computational cost, and they typically offer no guarantee that their solutions remain physically admissible.

By Denis Gueyffier (ONERA -- Institut Polytechnique de Paris)

arXiv Machine Learning

Jun 25

Silent Failures in Physics-Informed Neural Networks: Parameter Poisoning and the Limits of Loss-Based Validation

arXiv:2606. 25151v1 Announce Type: new Abstract: Physics-informed neural networks (PINNs) embed governing equations in their loss function, enabling mesh-free solutions to partial differential equations.

By David McShannon, Nicholas Dietrich

diffusion safety

arXiv AI

Aug 6

The Hamilton-Jacobi Theory of Deep Learning

arXiv:2605. 28983v2 Announce Type: replace-cross Abstract: In this paper, training a neural network is identified, exactly, as a search through Hamilton--Jacobi initial-value problems: each gradient step selects the initial data of a viscous Hamilton--Jacobi equation whose Hopf--Cole propagator best fits the observations; at inference, the input is the spatial point at which that solution is evaluated and the initial condition is already encoded in the weights.

By Jose Marie Antonio Mi\~noza, Erika Fille T. Legara, Christopher P. Monterola

llms safety

arXiv Machine Learning

Jun 24

Layer-wise Geometric Approximation Rates for Deep Networks

arXiv:2604. 20219v2 Announce Type: replace Abstract: Depth is widely viewed as a central contributor to the success of deep neural networks, whereas standard neural network approximation theory typically provides guarantees only for the final output and leaves the role of intermediate layers largely unclear.

By Shijun Zhang, Zuowei Shen, Yuesheng Xu

arXiv Machine Learning

Jun 11

Energy-Conserved Neural Pipelines: Attenuating Error Propagation in Modular Neural Networks via Physical Conservation Constraints

arXiv:2606. 11341v1 Announce Type: new Abstract: Modular neural network pipelines suffer from error compounding: noise at any module boundary propagates and potentially amplifies through subsequent modules.

By David Young, Swan Yi Htet

robotics safety

arXiv Machine Learning

Jul 17

Sharp Stability Threshold and Certification for Designing Stable Residual Architectures

arXiv:2607. 14576v1 Announce Type: new Abstract: We propose \emph{the sublinear-growth principle} for deep residual architectures -- a sharp stability threshold on the input-magnitude exponent of every residual block's velocity field: $$\|v(x, t)\| \leq c\,\|x\|^q + b, \qquad q \in [0, 1].

By Hyemin Gu, Michael Tyrrell, Tuhin Sahai, Markos A. Katsoulakis

Related stories

Hard Guarantees at a Measured Price: Entropy-Stable Learned Finite Volumes for Compressible Flow

Silent Failures in Physics-Informed Neural Networks: Parameter Poisoning and the Limits of Loss-Based Validation

The Hamilton-Jacobi Theory of Deep Learning

Layer-wise Geometric Approximation Rates for Deep Networks

Energy-Conserved Neural Pipelines: Attenuating Error Propagation in Modular Neural Networks via Physical Conservation Constraints

Sharp Stability Threshold and Certification for Designing Stable Residual Architectures