PE-CSNet: An equivariant network architecture with learnable patch-based sparse representation

Compressive sensing (CS) enables accurate signal reconstruction from sparse measurements and is widely applied in medical imaging, remote sensing, and image compression. However, designing an effective, task-specific sparse transform and the corresponding optimization procedure for high-quality CS remains challenging. This process typically requires expert domain knowledge and laborious parameter tuning. To address this issue, we present a Patch-based Equivariant deep unrolling architecture, termed PE-CSNet, for accurate CS recovery. While traditional CS methods generally use predefined patch-based transform sparsity, we generalize this idea by incorporating learnable transform sparsity that adapts to the specific CS task through an optimization-driven process. Specifically, we first establish a generalized patch-based CS model, which we solve via a block coordinate descent (BCD) algorithm. The BCD solver is then unrolled into a deep neural network, where all parameters of both the CS model and solver are learned through end-to-end training. To improve data efficiency, we introduce a stochastic equivariant training strategy that exploits the patch-wise structure of the network, enabling PE-CSNet to learn effectively even from limited data. We further provide a simpler, parameter-shared version of PE-CSNet and briefly discuss its convergence as an iterative solver. For practical applications, the network uses stage-specific (non-shared) parameters to enhance its expressive power and thereby improve its performance. On the tasks of CS magnetic resonance imaging (CS-MRI) and CS coded diffraction patterns (CS-CDP), PE-CSNet achieves state-of-the-art accuracy with fast computational speed, outperforming traditional methods and existing deep unrolling methods.

Emulating Cosmic Structure Formation with a Lagrangian Neural Cellular Automaton

Field-level inference of cosmological initial conditions from galaxy surveys requires a forward model that is simultaneously accurate in the non-linear regime, computationally efficient, and fully differentiable. Traditional N-body simulations are accurate but computationally prohibitive for iterative inference, while approximate solvers like Lagrangian Perturbation Theory (LPT) fail to capture the knotty halo-forming dynamics of the cosmic web at late times. We introduce the \textit{Lagrangian Neural Cellular Automaton} (LNCA), a hybrid deep learning framework that can be applied to emulate structure formation as a local, iterative dynamical process on a comoving lattice. Unlike standard Eulerian Convolutional Neural Networks (CNNs) which map fixed density fields, the LNCA operates in the Lagrangian frame, advecting the computational graph itself to follow the flow of mass. By training the network to learn only the \textit{residual} displacement corrections to the Zeldovich approximation, we achieve high-fidelity emulation of the non-linear physics while guaranteeing accuracy at large scales. We further constrain our model to produce complete trajectories, not just final states, by adopting an equivariant cellular automaton architecture, which recurrently iterates on its internal states to yield a dynamic history. The resulting model is strictly local, translationally and rotationally equivariant, and naturally supports continuous time integration, making it a reliable differentiable forward model for reconstructing the initial conditions of the universe from lightcone data. Our trained model supports percent-level precision in the power and cross spectra well into the non-linear regime ($k \lesssim 0.5 \, h \text{Mpc}^{-1}$), while requiring $\sim10^4$ times fewer learned parameters than comparable models which take the form of an interpretable internal dynamic rule set.

Scalable Peptide Design via Memory-Efficient Equivariant Transformer

Target-specific peptide design requires sequence and structure co-design under full atom geometric constraints. Latent generative frameworks offer an effective route for this problem by compressing fine grained atomic structures into block level latent representations and performing conditional generation in a compact latent space. However, the scalability of such systems depends heavily on the geometric backbone used throughout their encoding, decoding, and denoising components. We introduce MEET (Memory Efficient Equivariant Transformer), an E(3) equivariant backbone for scalable atomistic peptide modeling. MEET maintains coupled invariant scalar and equivariant vector feature streams, while reformulating geometric computation around memory efficient attention. It initializes vector features through global coordinate aggregation, incorporates pairwise distances through augmented query and key dot products, and injects covalent bond information through sparse bond adaptation. Integrated into a VAE and latent diffusion pipeline for full atom peptide generation, MEET achieves linear memory scaling with atom count and improves generation quality over existing peptide design methods. Experiments on large scale AFDB derived datasets further show that the proposed backbone supports systematic model and data scaling, leading to better binding affinity, physical validity, and sample diversity.