Supervised machine learning (ML) methods are emerging as valid alternatives to standard mathematical methods for identifying knots in long, collapsed polymers. Here, we introduce a hybrid supervised/unsupervised ML approach for knot classification based on a variational autoencoder enhanced with a knot type classifier (VAEC). The neat organization of knots in its latent representation suggests that the VAEC, only based on an arbitrary labeling of three-dimensional configurations, has grasped complex topological concepts such as chirality, unknotting number, braid index, and the grouping in families such as achiral, torus, and twist knots. The understanding of topological concepts is confirmed by the ability of the VAEC to distinguish the chirality of knots $9_{42}$ and $10_{71}$ not used for its training and with a notoriously undetected chirality to standard tools. The well-organized latent space is also key for generating configurations with the decoder that reliably preserves the topology of the input ones. Our findings demonstrate the ability of a hybrid supervised-generative ML algorithm to capture different topological features of entangled filaments and to exploit this knowledge to faithfully reconstruct or produce new knotted configurations without simulations.

We thank all reviewers for their comments. A very solid/extensive experimental validation is performed (R1&R2&R3). Moreover, the paper is well written, easy to follow (R1&R2) and well-organized (R3). In the following we address all comments individually. Novelty of the technique to achieve equivariance.

In this study, we classify the magnetic chirality of solar filaments from H-Alpha observations using state-of-the-art image classification models. We establish the first reproducible baseline for solar filament chirality classification on the MAGFiLO dataset. The MAGFiLO dataset contains over 10,000 manually-annotated filaments from GONG H-Alpha observations, making it the largest dataset for filament detection and classification to date. Prior studies relied on much smaller datasets, which limited their generalizability and comparability. We fine-tuned several pre-trained, image classification architectures, including ResNet, WideResNet, ResNeXt, and ConvNeXt, and also applied data augmentation and per-class loss weights to optimize the models. Our best model, ConvNeXtBase, achieves a per-class accuracy of 0.69 for left chirality filaments and $0.73$ for right chirality filaments.

Institute of Materials Chemistry, TU Wien, A-1060 Vienna, Austria Conditional Flow Matching(CFM) represents a fast and high-quality approach to generative modelling, but in many applications it is of interest to steer the generated samples towards precise requirements. While steering approaches like gradient-based guidance, sequential Monte Carlo steering or Feynman-Kac steering are well established for diffusion models, they have not been extended to flow matching approaches yet. In this work, we formulate this requirement as tilting the output with an energy potential. We derive, for the first time, Feynman-Kac steering for CFM. We evaluate our approach on a set of synthetic tasks, including the generation of tilted distributions in a high-dimensional space, which is a particularly challenging case for steering approaches. We then demonstrate the impact of Feynman-Kac steered CFM on the previously unsolved challenge of generated transition states of chemical reactions with the correct chirality, where the reactants or products can have a different handedness, leading to geometric constraints of the viable reaction pathways connecting reactants and products. Code to reproduce this study is avaiable open-source at https://github.com/heid-lab/fkflow. I. INTRODUCTION Since its introduction by Lipman et al. [1], Conditional Flow Matching (CFM) has seen several interesting applications, ranging from image [1], audio [2] and video [3] generation to decision-making [4], time series modelling [5], protein modelling [6, 7] or molecular structure design [8], amongst others. CFM transforms samples from a source distribution (such as random noise) to samples following a given target distribution (such as images or molecular structures) by modelling probability paths via vector fields. It largely improves on diffusion-based methods both in quality and speed, establishing CFM as a popular generative method [1].

We introduce Chi-Geometry - a library that generates graph data for testing and benchmarking GNNs' ability to predict chirality. Chi-Geometry generates synthetic graph samples with (i) user-specified geometric and topological traits to isolate certain types of samples and (ii) randomized node positions and species to minimize extraneous correlations. Each generated graph contains exactly one chiral center labeled either R or S, while all other nodes are labeled N/A (non-chiral). The generated samples are then combined into a cohesive dataset that can be used to assess a GNN's ability to predict chirality as a node classification task. Chi-Geometry allows more interpretable and less confounding benchmarking of GNNs for prediction of chirality in the graph samples which can guide the design of new GNN architectures with improved predictive performance. We illustrate Chi-Geometry's efficacy by using it to generate synthetic datasets for benchmarking various state-of-the-art (SOTA) GNN architectures. The conclusions of these benchmarking results guided our design of two new GNN architectures. The first GNN architecture established all-to-all connections in the graph to accurately predict chirality across all challenging configurations where previously tested SOTA models failed, but at a computational cost (both for training and inference) that grows quadratically with the number of graph nodes. The second GNN architecture avoids all-to-all connections by introducing a virtual node in the original graph structure of the data, which restores the linear scaling of training and inference computational cost with respect to the number of nodes in the graph, while still ensuring competitive accuracy in detecting chirality with respect to SOTA GNN architectures.

Frustrated itinerant magnets often exhibit complex noncollinear or noncoplanar magnetic orders which support topological electronic structures. A canonical example is the anomalous quantum Hall state with a chiral spin order stabilized by electron-spin interactions on a triangular lattice. While a long-range magnetic order cannot survive thermal fluctuations in two dimensions, the chiral order which results from the breaking of a discrete Ising symmetry persists even at finite temperatures. We present a scalable machine learning (ML) framework to model the complex electron-mediated spin-spin interactions that stabilize the chiral magnetic domains in a triangular lattice. Large-scale dynamical simulations, enabled by the ML force-field models, are performed to investigate the coarsening of chiral domains after a thermal quench. While the chiral phase is described by a broken $Z_2$ Ising-type symmetry, we find that the characteristic size of chiral domains increases linearly with time, in stark contrast to the expected Allen-Cahn domain growth law for a non-conserved Ising order parameter field. The linear growth of the chiral domains is attributed to the orientational anisotropy of domain boundaries. Our work also demonstrates the promising potential of ML models for large-scale spin dynamics of itinerant magnets.

The prediction of chemical reactions has gained significant interest within the machine learning community in recent years, owing to its complexity and crucial applications in chemistry. However, model evaluation for this task has been mostly limited to simple metrics like top-k accuracy, which obfuscates fine details of a model's limitations. Inspired by progress in other fields, we propose a new assessment scheme that builds on top of current approaches, steering towards a more holistic evaluation. We introduce the following key components for this goal: CHORISO, a curated dataset along with multiple tailored splits to recreate chemically relevant scenarios, and a collection of metrics that provide a holistic view of a model's advantages and limitations. Application of this method to state-of-the-art models reveals important differences on sensitive fronts, especially stereoselectivity and chemical out-of-distribution generalization. Our work paves the way towards robust prediction models that can ultimately accelerate chemical discovery.

In this study, we conduct a comprehensive theoretical analysis of a Fibonacci quasicrystalline stacking of ferromagnetic layers, potentially realizable using van der Waals magnetic materials. We construct a model of this magnetic heterostructure, which includes up to second neighbor interlayer magnetic interactions, that displays a complex relationship between geometric frustration and magnetic order in this quasicrystalline system. To navigate the parameter space and identify distinct magnetic phases, we employ a machine learning approach, which proves to be a powerful tool in revealing the complex magnetic behavior of this system. We offer a thorough description of the magnetic phase diagram as a function of the model parameters. Notably, we discover among other collinear and non-collinear phases, a unique ferromagnetic alternating helical phase. In this non-collinear quasiperiodic ferromagnetic configuration the magnetization decreases logarithmically with the stack height.

Graph Neural Networks (GNNs) play a fundamental role in many deep learning problems, in particular in cheminformatics. However, typical GNNs cannot capture the concept of chirality, which means they do not distinguish between the 3D graph of a chemical compound and its mirror image (enantiomer). The ability to distinguish between enantiomers is important especially in drug discovery because enantiomers can have very distinct biochemical properties. In this paper, we propose a theoretically justified message-passing scheme, which makes GNNs sensitive to the order of node neighbors. We apply that general concept in the context of molecular chirality to construct Chiral Edge Neural Network (ChiENN) layer which can be appended to any GNN model to enable chirality-awareness. Our experiments show that adding ChiENN layers to a GNN outperforms current state-of-the-art methods in chiral-sensitive molecular property prediction tasks.

Molecules are frequently represented as graphs, but the underlying 3D molecular geometry (the locations of the atoms) ultimately determines most molecular properties. However, most molecules are not static and at room temperature adopt a wide variety of geometries or $\textit{conformations}$. The resulting distribution on geometries $p(x)$ is known as the Boltzmann distribution, and many molecular properties are expectations computed under this distribution. Generating accurate samples from the Boltzmann distribution is therefore essential for computing these expectations accurately. Traditional sampling-based methods are computationally expensive, and most recent machine learning-based methods have focused on identifying $\textit{modes}$ in this distribution rather than generating true $\textit{samples}$. Generating such samples requires capturing conformational variability, and it has been widely recognized that the majority of conformational variability in molecules arises from rotatable bonds. In this work, we present VonMisesNet, a new graph neural network that captures conformational variability via a variational approximation of rotatable bond torsion angles as a mixture of von Mises distributions. We demonstrate that VonMisesNet can generate conformations for arbitrary molecules in a way that is both physically accurate with respect to the Boltzmann distribution and orders of magnitude faster than existing sampling methods.