Flow Matching for Geometric Trajectory Simulation

The simulation of N-body systems is a fundamental problem with applications in a wide range of fields, such as molecular dynamics, biochemistry, and pedestrian dynamics. Machine learning has become an invaluable tool for scaling physics-based simulators and developing models directly from experimental data. In particular, recent advances based on deep generative modeling and geometric deep learning have enabled probabilistic simulation by modeling complex distributions over trajectories while respecting the permutation symmetry that is fundamental to N-body systems. However, to generate realistic trajectories, existing methods must learn complex transformations starting from uninformed noise and do not allow for the exploitation of domain-informed priors. In this work, we propose STFlow to address this limitation. By leveraging flow matching and data-dependent couplings, STFlow facilitates physics-informed simulation of geometric trajectories without sacrificing model expressivity or scalability. Our evaluation on N-body dynamical systems, molecular dynamics, and pedestrian dynamics benchmarks shows that STFlow produces significantly lower prediction errors while enabling more efficient inference, highlighting the benefits of employing physics-informed prior distributions in probabilistic geometric trajectory modeling.

Plasma State Monitoring and Disruption Characterization using Multimodal VAEs

When a plasma disrupts in a tokamak, significant heat and electromagnetic loads are deposited onto the surrounding device components. These forces scale with plasma current and magnetic field strength, making disruptions one of the key challenges for future devices. Unfortunately, disruptions are not fully understood, with many different underlying causes that are difficult to anticipate. Data-driven models have shown success in predicting them, but they only provide limited interpretability. On the other hand, large-scale statistical analyses have been a great asset to understanding disruptive patterns. In this paper, we leverage data-driven methods to find an interpretable representation of the plasma state for disruption characterization. Specifically, we use a latent variable model to represent diagnostic measurements as a low-dimensional, latent representation. We build upon the Variational Autoencoder (VAE) framework, and extend it for (1) continuous projections of plasma trajectories; (2) a multimodal structure to separate operating regimes; and (3) separation with respect to disruptive regimes. Subsequently, we can identify continuous indicators for the disruption rate and the disruptivity based on statistical properties of measurement data. The proposed method is demonstrated using a dataset of approximately 1600 TCV discharges, selecting for flat-top disruptions or regular terminations. We evaluate the method with respect to (1) the identified disruption risk and its correlation with other plasma properties; (2) the ability to distinguish different types of disruptions; and (3) downstream analyses. For the latter, we conduct a demonstrative study on identifying parameters connected to disruptions using counterfactual-like analysis. Overall, the method can adequately identify distinct operating regimes characterized by varying proximity to disruptions in an interpretable manner.

Robust Confinement State Classification with Uncertainty Quantification through Ensembled Data-Driven Methods

Maximizing fusion performance in tokamaks relies on high energy confinement, often achieved through distinct operating regimes. The automated labeling of these confinement states is crucial to enable large-scale analyses or for real-time control applications. While this task becomes difficult to automate near state transitions or in marginal scenarios, much success has been achieved with data-driven models. However, these methods generally provide predictions as point estimates, and cannot adequately deal with missing and/or broken input signals. To enable wide-range applicability, we develop methods for confinement state classification with uncertainty quantification and model robustness. We focus on off-line analysis for TCV discharges, distinguishing L-mode, H-mode, and an in-between dithering phase (D). We propose ensembling data-driven methods on two axes: model formulations and feature sets. The former considers a dynamic formulation based on a recurrent Fourier Neural Operator-architecture and a static formulation based on gradient-boosted decision trees. These models are trained using multiple feature groupings categorized by diagnostic system or physical quantity. A dataset of 302 TCV discharges is fully labeled, and will be publicly released. We evaluate our method quantitatively using Cohen's kappa coefficient for predictive performance and the Expected Calibration Error for the uncertainty calibration. Furthermore, we discuss performance using a variety of common and alternative scenarios, the performance of individual components, out-of-distribution performance, cases of broken or missing signals, and evaluate conditionally-averaged behavior around different state transitions. Overall, the proposed method can distinguish L, D and H-mode with high performance, can cope with missing or broken signals, and provides meaningful uncertainty estimates.

Deep Neural Cellular Potts Models

The cellular Potts model (CPM) is a powerful computational method for simulating collective spatiotemporal dynamics of biological cells. To drive the dynamics, CPMs rely on physics-inspired Hamiltonians. However, as first principles remain elusive in biology, these Hamiltonians only approximate the full complexity of real multicellular systems. To address this limitation, we propose NeuralCPM, a more expressive cellular Potts model that can be trained directly on observational data. At the core of NeuralCPM lies the Neural Hamiltonian, a neural network architecture that respects universal symmetries in collective cellular dynamics. Moreover, this approach enables seamless integration of domain knowledge by combining known biological mechanisms and the expressive Neural Hamiltonian into a hybrid model. Our evaluation with synthetic and real-world multicellular systems demonstrates that NeuralCPM is able to model cellular dynamics that cannot be accounted for by traditional analytical Hamiltonians.

Aspect-Based Few-Shot Learning

We generalize the formulation of few-shot learning by introducing the concept of an aspect. In the traditional formulation of few-shot learning, there is an underlying assumption that a single "true" label defines the content of each data point. This label serves as a basis for the comparison between the query object and the objects in the support set. However, when a human expert is asked to execute the same task without a predefined set of labels, they typically consider the rest of the data points in the support set as context. This context specifies the level of abstraction and the aspect from which the comparison can be made. In this work, we introduce a novel architecture and training procedure that develops a context given the query and support set and implements aspect-based few-shot learning that is not limited to a predetermined set of classes. We demonstrate that our method is capable of forming and using an aspect for few-shot learning on the Geometric Shapes and Sprites dataset. The results validate the feasibility of our approach compared to traditional few-shot learning.

Understanding complex crowd dynamics with generative neural simulators

Understanding the dynamics of pedestrian crowds is an outstanding challenge crucial for designing efficient urban infrastructure and ensuring safe crowd management. To this end, both small-scale laboratory and large-scale real-world measurements have been used. However, these approaches respectively lack statistical resolution and parametric controllability, both essential to discovering physical relationships underlying the complex stochastic dynamics of crowds. Here, we establish an investigation paradigm that offers laboratory-like controllability, while ensuring the statistical resolution of large-scale real-world datasets. Using our data-driven Neural Crowd Simulator (NeCS), which we train on large-scale data and validate against key statistical features of crowd dynamics, we show that we can perform effective surrogate crowd dynamics experiments without training on specific scenarios. We not only reproduce known experimental results on pairwise avoidance, but also uncover the vision-guided and topological nature of N-body interactions. These findings show how virtual experiments based on neural simulation enable data-driven scientific discovery.

Topological degree as a discrete diagnostic for disentanglement, with applications to the $Δ$VAE

We investigate the ability of Diffusion Variational Autoencoder ($\Delta$VAE) with unit sphere $\mathcal{S}^2$ as latent space to capture topological and geometrical structure and disentangle latent factors in datasets. For this, we introduce a new diagnostic of disentanglement: namely the topological degree of the encoder, which is a map from the data manifold to the latent space. By using tools from homology theory, we derive and implement an algorithm that computes this degree. We use the algorithm to compute the degree of the encoder of models that result from the training procedure. Our experimental results show that the $\Delta$VAE achieves relatively small LSBD scores, and that regardless of the degree after initialization, the degree of the encoder after training becomes $-1$ or $+1$, which implies that the resulting encoder is at least homotopic to a homeomorphism.

Accelerating Simulation of Two-Phase Flows with Neural PDE Surrogates

Simulation is a powerful tool to better understand physical systems, but generally requires computationally expensive numerical methods. Downstream applications of such simulations can become computationally infeasible if they require many forward solves, for example in the case of inverse design with many degrees of freedom. In this work, we investigate and extend neural PDE solvers as a tool to aid in scaling simulations for two-phase flow problems, and simulations of oil expulsion from a pore specifically. We extend existing numerical methods for this problem to a more complex setting involving varying geometries of the domain to generate a challenging dataset. Further, we investigate three prominent neural PDE solver methods, namely the UNet, DRN and U-FNO, and extend them for characteristics of the oil-expulsion problem: (1) spatial conditioning on the geometry; (2) periodicity in the boundary; (3) approximate mass conservation. We scale all methods and benchmark their speed-accuracy trade-off, evaluate qualitative properties, and perform an ablation study. We find that the investigated methods can accurately model the droplet dynamics with up to three orders of magnitude speed-up, that our extensions improve performance over the baselines, and that the introduced varying geometries constitute a significantly more challenging setting over the previously considered oil expulsion problem.

Efficient Probabilistic Modeling of Crystallization at Mesoscopic Scale

Crystallization processes at the mesoscopic scale, where faceted, dendritic growth, and multigrain formation can be observed, are of particular interest within materials science and metallurgy. These processes are highly nonlinear, stochastic, and sensitive to small perturbations of system parameters and initial conditions. Methods for the simulation of these processes have been developed using discrete numerical models, but these are computationally expensive. This work aims to scale crystal growth simulation with a machine learning emulator. Specifically, autoregressive latent variable models are well suited for modeling the joint distribution over system parameters and the crystallization trajectories. However, successfully training such models is challenging due to the stochasticity and sensitivity of the system. Existing approaches consequently fail to produce diverse and faithful crystallization trajectories. In this paper, we introduce the Crystal Growth Neural Emulator (CGNE), a probabilistic model for efficient crystal growth emulation at the mesoscopic scale that overcomes these challenges. We validate CGNE results using the morphological properties of the crystals produced by numerical simulation. CGNE delivers a factor of 11 improvement in inference time and performance gains compared with recent state-of-the-art probabilistic models for dynamical systems.

Similarity Equivariant Graph Neural Networks for Homogenization of Metamaterials

Soft, porous mechanical metamaterials exhibit pattern transformations that may have important applications in soft robotics, sound reduction and biomedicine. To design these innovative materials, it is important to be able to simulate them accurately and quickly, in order to tune their mechanical properties. Since conventional simulations using the finite element method entail a high computational cost, in this article we aim to develop a machine learning-based approach that scales favorably to serve as a surrogate model. To ensure that the model is also able to handle various microstructures, including those not encountered during training, we include the microstructure as part of the network input. Therefore, we introduce a graph neural network that predicts global quantities (energy, stress stiffness) as well as the pattern transformations that occur (the kinematics). To make our model as accurate and data-efficient as possible, various symmetries are incorporated into the model. The starting point is an E(n)-equivariant graph neural network (which respects translation, rotation and reflection) that has periodic boundary conditions (i.e., it is in-/equivariant with respect to the choice of RVE), is scale in-/equivariant, can simulate large deformations, and can predict scalars, vectors as well as second and fourth order tensors (specifically energy, stress and stiffness). The incorporation of scale equivariance makes the model equivariant with respect to the similarities group, of which the Euclidean group E(n) is a subgroup. We show that this network is more accurate and data-efficient than graph neural networks with fewer symmetries. To create an efficient graph representation of the finite element discretization, we use only the internal geometrical hole boundaries from the finite element mesh to achieve a better speed-up and scaling with the mesh size.