The warping estimation module W is based on an hourglass with five conv3 3 - bn - relu - pool2 2 in the encoders and five upsample2 2 - conv3 3 - bn - relu blocks in the decoders. In G, we use the Johnson architecture [ 3 ] with two down-sampling blocks, six residual-blocks and two up-sampling blocks. The design follows [ 7 ]. The inputs are the base image, displacement field, and inpainting map. It downsampled 4 and upsampled 4 to get the output, i.e. the reconstructed image.

Recent research has observed that in machine learning optimization, gradient descent (GD) often operates at the edge of stability (EoS) [Cohen et al., 2021], where the stepsizes are set to be large, resulting in non-monotonic losses induced by the GD iterates. This paper studies the convergence and implicit bias of constant-stepsize GD for logistic regression on linearly separable data in the EoS regime. Despite the presence of local oscillations, we prove that the logistic loss can be minimized by GD with any constant stepsize over a long time scale. Furthermore, we prove that with any constant stepsize, the GD iterates tend to infinity when projected to a max-margin direction (the hard-margin SVM direction) and converge to a fixed vector that minimizes a strongly convex potential when projected to the orthogonal complement of the max-margin direction. In contrast, we also show that in the EoS regime, GD iterates may diverge catastrophically under the exponential loss, highlighting the superiority of the logistic loss. These theoretical findings are in line with numerical simulations and complement existing theories on the convergence and implicit bias of GD for logistic regression, which are only applicable when the stepsizes are sufficiently small.

We propose a complement to constitutive modeling that augments neural networks with material principles to capture anisotropy and inelasticity at finite strains. The key element is a dual potential that governs dissipation, consistently incorporates anisotropy, and-unlike conventional convex formulations-satisfies the dissipation inequality without requiring convexity. Our neural network architecture employs invariant-based input representations in terms of mixed elastic, inelastic and structural tensors. It adapts Input Convex Neural Networks, and introduces Input Monotonic Neural Networks to broaden the admissible potential class. To bypass exponential-map time integration in the finite strain regime and stabilize the training of inelastic materials, we employ recurrent Liquid Neural Networks. The approach is evaluated at both material point and structural scales. We benchmark against recurrent models without physical constraints and validate predictions of deformation and reaction forces for unseen boundary value problems. In all cases, the method delivers accurate and stable performance beyond the training regime. The neural network and finite element implementations are available as open-source and are accessible to the public via https://doi.org/10.5281/zenodo.17199965.

Experiments demonstrate the advantage of our framework.

This problem is challenging, because KGs can be massive and incomplete.

To show Proposition 1, we need the following defition and lemma. Suppose that K is a proper cone. A sector-cone is always axially symmetric. Therefore, when C is convex, it is axially symmetric. Therefore, a sector-cone is always axially symmetric.

We thank reviewers R1, R2, R3, and R4 for their constructive and helpful feedback. We aim to explore these ideas in future work. Our work is meant to complement these previous studies. We modified the Related work section to discuss and point out how these proposals complement our work. " This is not a paper searching for state of the art results, and it should not be We added these results to the revised manuscript.

The general form for an exponential family likelihood is still retained. The prior-to-posterior conversion simply involves updating the prior parameters with sufficient statistics from the data. The inequality is by Markov's inequality. This concludes the proof.C.1 Proof of Theorem 1 Since we have context generated by some random process, we instead turn to martingales. We see that the choice of action given observed context depends on past rounds.