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.
arXiv:2602. 18849v2 Announce Type: replace-cross Abstract: We develop a sensitivity analysis for transformer attention in a geometry aligned with tokenwise computation.
arXiv:2607. 04993v1 Announce Type: cross Abstract: Many phenomena of deep learning are dynamical: they concern not only which minima exist, but how gradient descent reaches, avoids, or selects among them.
arXiv:2507. 05164v2 Announce Type: replace-cross Abstract: In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms.
arXiv:2605. 01288v3 Announce Type: replace Abstract: In deep networks with small initialization, training exhibits long plateaus separated by sharp feature-acquisition transitions.
arXiv:2606. 26705v1 Announce Type: cross Abstract: Feedforward neural network (NN) expressivity is typically studied by emulating optimal basis-expansion schemes.
arXiv:2509. 16395v2 Announce Type: replace-cross Abstract: Evolutionary deep neural networks (EDNNs) solve time-dependent partial differential equations by evolving the neural-network parameters sequentially in time through a local least-squares problem.
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].
arXiv:2607. 02194v1 Announce Type: new Abstract: Physics-informed neural networks (PINNs) have emerged as a promising route to solve partial differential equations, yet they have struggled to reach the precision of classical solvers.
arXiv:2505. 06589v2 Announce Type: replace-cross Abstract: Modern machine learning repeatedly manipulates probability measures: empirical datasets, generated samples, latent distributions, class-conditional laws, particle systems, weights of wide networks and attention patterns.
arXiv:2606. 04420v1 Announce Type: new Abstract: Physics-informed neural networks (PINNs) approximate solutions of ODEs and PDEs by minimising a weighted combination of residual, boundary, initial, and data losses.
arXiv:2607. 13380v1 Announce Type: new Abstract: Predictive Coding (PC) offers a biologically motivated alternative to backpropagation via local weight updates, yet routing error between layers still relies on an autograd Jacobian-transpose ($J^\top$) product - the last non-local operation in PC.
arXiv:2506. 08764v3 Announce Type: replace Abstract: Deep neural networks are known to suffer from exploding or vanishing gradients as depth increases, a phenomenon closely tied to the spectral behavior of the input-output Jacobian.