G.1 Monoids Monoids are one of the simplest algebraic structure.

Recent advances in implicit neural representations show great promise when it comes to generating numerical solutions to partial differential equations. Compared to conventional alternatives, such representations employ parameterized neural networks to define, in a mesh-free manner, signals that are highly-detailed, continuous, and fully differentiable. In this work, we present a novel machine learning approach for topology optimization--an important class of inverse problems with high-dimensional parameter spaces and highly nonlinear objective landscapes. To effectively leverage neural representations in the context of mesh-free topology optimization, we use multilayer perceptrons to parameterize both density and displacement fields. Our experiments indicate that our method is highly competitive for minimizing structural compliance objectives, and it enables self-supervised learning of continuous solution spaces for topology optimization problems.

Large vision-language models (LVLMs) have recently achieved rapid progress, exhibiting great perception and reasoning abilities concerning visual information. However, when faced with prompts in different sizes of solution spaces, LVLMs fail to always give consistent answers regarding the same knowledge point. This inconsistency of answers between different solution spaces is prevalent in LVLMs and erodes trust. To this end, we provide a multi-modal benchmark ConBench, to intuitively analyze how LVLMs perform when the solution space of a prompt revolves around a knowledge point. Based on the ConBench tool, we are the first to reveal the tapestry and get the following findings: (1) In the discriminate realm, the larger the solution space of the prompt, the lower the accuracy of the answers.

Deep neural networks are starting to show their potential for solving partial differential equations (PDEs)inavarietyofproblemdomains,includingturbulentflow,heattransfer,elastodynamics,and many more [1, 2, 3, 4, 5]. Thanks to their smooth and analytically-differentiable nature, implicit neural representations with periodic activation functions are emerging as a particularly attractive and powerful option in this context [4].

Combinatorial optimization (CO) problems are often NP-hard and thus out of reach for exact algorithms, making them a tempting domain to apply machine learning methods. The highly structured constraints in these problems can hinder either optimization or sampling directly in the solution space.On the other hand, GFlowNets have recently emerged as a powerful machinery to efficiently sample from composite unnormalized densities sequentially and have the potential to amortize such solution-searching processes in CO, as well as generate diverse solution candidates.In this paper, we design Markov decision processes (MDPs) for different combinatorial problems and propose to train conditional GFlowNets to sample from the solution space.

Federated reinforcement learning (FRL) enables distributed learning of optimal policies while preserving local data privacy through gradient sharing.However, FRL faces the risk of data privacy leaks, where attackers exploit shared gradients to reconstruct local training data.Compared to traditional supervised federated learning, successful reconstruction in FRL requires the generated data not only to match the shared gradients but also to align with real transition dynamics of the environment (i.e., aligning with the real data transition distribution).To address this issue, we propose a novel attack method called Regularization Gradient Inversion Attack (RGIA), which enforces prior-knowledge-based regularization on states, rewards, and transition dynamics during the optimization process to ensure that the reconstructed data remain close to the true transition distribution.Theoretically, we prove that the prior-knowledge-based regularization term narrows the solution space from a broad set containing spurious solutions to a constrained subset that satisfies both gradient matching and true transition dynamics.Extensive experiments on control tasks and autonomous driving tasks demonstrate that RGIA can effectively constrain reconstructed data transition distributions and thus successfully reconstruct local private data.