We introduce a general, scalable computational framework for multi-axis 3D printing based on implicit neural fields (INFs) that unifies all stages of toolpath generation and global collision-free motion planning. In our pipeline, input models are represented as signed distance fields, with fabrication objectives such as support-free printing, surface finish quality, and extrusion control being directly encoded in the optimization of an implicit guidance field. This unified approach enables toolpath optimization across both surface and interior domains, allowing shell and infill paths to be generated via implicit field interpolation. The printing sequence and multi-axis motion are then jointly optimized over a continuous quaternion field. Our continuous formulation constructs the evolving printing object as a time-varying SDF, supporting differentiable global collision handling throughout INF-based motion planning. Compared to explicit-representation-based methods, INF-3DP achieves up to two orders of magnitude speedup and significantly reduces waypoint-to-surface error. We validate our framework on diverse, complex models and demonstrate its efficiency with physical fabrication experiments using a robot-assisted multi-axis system.

The Rust programming language is an attractive choice for robotics and related fields, offering highly efficient and memory-safe code. However, a key limitation preventing its broader adoption in these domains is the lack of high-quality, well-supported Automatic Differentiation (AD)-a fundamental technique that enables convenient derivative computation by systematically accumulating data during function evaluation. In this work, we introduce ad-trait, a new Rust-based AD library. Our implementation overloads Rust's standard floating-point type with a flexible trait that can efficiently accumulate necessary information for derivative computation. The library supports both forward-mode and reverse-mode automatic differentiation, making it the first operator-overloading AD implementation in Rust to offer both options. Additionally, ad-trait leverages Rust's performance-oriented features, such as Single Instruction, Multiple Data acceleration in forward-mode AD, to enhance efficiency. Through benchmarking experiments, we show that our library is among the fastest AD implementations across several programming languages for computing derivatives. Moreover, it is already integrated into a Rust-based robotics library, where we showcase its ability to facilitate fast optimization procedures. We conclude with a discussion of the limitations and broader implications of our work.

The identification of the Purkinje conduction system in the heart is a challenging task, yet essential for a correct definition of cardiac digital twins for precision cardiology. Here, we propose a probabilistic approach for identifying the Purkinje network from non-invasive clinical data such as the standard electrocardiogram (ECG). We use cardiac imaging to build an anatomically accurate model of the ventricles; we algorithmically generate a rule-based Purkinje network tailored to the anatomy; we simulate physiological electrocardiograms with a fast model; we identify the geometrical and electrical parameters of the Purkinje-ECG model with Bayesian optimization and approximate Bayesian computation. The proposed approach is inherently probabilistic and generates a population of plausible Purkinje networks, all fitting the ECG within a given tolerance. In this way, we can estimate the uncertainty of the parameters, thus providing reliable predictions. We test our methodology in physiological and pathological scenarios, showing that we are able to accurately recover the ECG with our model. We propagate the uncertainty in the Purkinje network parameters in a simulation of conduction system pacing therapy. Our methodology is a step forward in creation of digital twins from non-invasive data in precision medicine. An open source implementation can be found at http://github.com/fsahli/purkinje-learning

This paper investigates how representation learning can enable optimal control in unknown and complex dynamics, such as chaotic and non-linear systems, without relying on prior domain knowledge of the dynamics. The core idea is to establish an equivariant geometry that is diffeomorphic to the manifold defined by a dynamical system and to perform optimal control within this corresponding geometry, which is a non-trivial task. To address this challenge, Koopman Embed to Equivariant Control (KEEC) is proposed for model learning and control. Inspired by Lie theory, KEEC begins by learning a non-linear dynamical system defined on a manifold and embedding trajectories into a Lie group. Subsequently, KEEC formulates an equivariant value function equation in reinforcement learning on the equivariant geometry, ensuring an invariant effect as the value function on the original manifold. By deriving analytical-form optimal actions on the equivariant value function, KEEC theoretically achieves quadratic convergence for the optimal equivariant value function by leveraging the differential information on the equivariant geometry. The effectiveness of KEEC is demonstrated in challenging dynamical systems, including chaotic ones like Lorenz-63. Notably, our results show that isometric functions, which maintain the compactness and completeness of geometry while preserving metric and differential information, consistently outperform loss functions lacking these characteristics.

Machine learning approaches for solving partial differential equations require learning mappings between function spaces. While convolutional or graph neural networks are constrained to discretized functions, neural operators present a promising milestone toward mapping functions directly. Despite impressive results they still face challenges with respect to the domain geometry and typically rely on some form of discretization. In order to alleviate such limitations, we present CORAL, a new method that leverages coordinate-based networks for solving PDEs on general geometries. CORAL is designed to remove constraints on the input mesh, making it applicable to any spatial sampling and geometry. Its ability extends to diverse problem domains, including PDE solving, spatio-temporal forecasting, and inverse problems like geometric design. CORAL demonstrates robust performance across multiple resolutions and performs well in both convex and non-convex domains, surpassing or performing on par with state-of-the-art models.

Various applications ranging from robotics to climate science require modeling signals on non-Euclidean domains, such as the sphere. Gaussian process models on manifolds have recently been proposed for such tasks, in particular when uncertainty quantification is needed. In the manifold setting, vector-valued signals can behave very differently from scalar-valued ones, with much of the progress so far focused on modeling the latter. The former, however, are crucial for many applications, such as modeling wind speeds or force fields of unknown dynamical systems. In this paper, we propose novel Gaussian process models for vector-valued signals on manifolds that are intrinsically defined and account for the geometry of the space in consideration. We provide computational primitives needed to deploy the resulting Hodge-Mat\'ern Gaussian vector fields on the two-dimensional sphere and the hypertori. Further, we highlight two generalization directions: discrete two-dimensional meshes and "ideal" manifolds like hyperspheres, Lie groups, and homogeneous spaces. Finally, we show that our Gaussian vector fields constitute considerably more refined inductive biases than the extrinsic fields proposed before.

Data assimilation is crucial in a wide range of applications, but it often faces challenges such as high computational costs due to data dimensionality and incomplete understanding of underlying mechanisms. To address these challenges, this study presents a novel assimilation framework, termed Latent Assimilation with Implicit Neural Representations (LAINR). By introducing Spherical Implicit Neural Representations (SINR) along with a data-driven uncertainty estimator of the trained neural networks, LAINR enhances efficiency in assimilation process. Experimental results indicate that LAINR holds certain advantage over existing methods based on AutoEncoders, both in terms of accuracy and efficiency.

An SGM involves a Score-based generative models (SGMs) learn a stochastic forward and backward process. In the forward family of noise-conditional score functions corresponding process, also known as the diffusion process, noise with to the data density perturbed with gradually increasing variances is added to each data point increasingly large amounts of noise. These until the original structure is lost, transforming data into perturbed data densities are linked together by pure noise. The backward process attempts to reverse the the Fokker-Planck equation (FPE), a partial differential diffusion process by using a neural network (called a noiseconditional equation (PDE) governing the spatialtemporal score model) that is trained to gradually denoise evolution of a density undergoing a diffusion the data, effectively transforming pure noise into clean data process. In this work, we derive a corresponding samples. The neural network is trained with a denoising equation called the score FPE that score matching objective (Hyvärinen & Dayan, 2005; Vincent, characterizes the noise-conditional scores of the 2011) to estimate the score (i.e., the gradient of the perturbed data densities (i.e., their gradients). Surprisingly, log-likelihood function) of the data density perturbed with despite the impressive empirical performance, various amounts of noise (as in forward process).

Existing Score-Based Models (SBMs) can be categorized into constrained SBMs (CSBMs) or unconstrained SBMs (USBMs) according to their parameterization approaches. CSBMs model probability density functions as Boltzmann distributions, and assign their predictions as the negative gradients of some scalar-valued energy functions. On the other hand, USBMs employ flexible architectures capable of directly estimating scores without the need to explicitly model energy functions. In this paper, we demonstrate that the architectural constraints of CSBMs may limit their modeling ability. In addition, we show that USBMs' inability to preserve the property of conservativeness may lead to degraded performance in practice. To address the above issues, we propose Quasi-Conservative Score-Based Models (QCSBMs) for keeping the advantages of both CSBMs and USBMs. Our theoretical derivations demonstrate that the training objective of QCSBMs can be efficiently integrated into the training processes by leveraging the Hutchinson's trace estimator. In addition, our experimental results on the CIFAR-10, CIFAR-100, ImageNet, and SVHN datasets validate the effectiveness of QCSBMs. Finally, we justify the advantage of QCSBMs using an example of a one-layered autoencoder.

Enriching the quality of early childhood education with interactive math learning at home systems, empowered by recent advances in conversational AI technologies, is slowly becoming a reality. With this motivation, we implement a multimodal dialogue system to support play-based learning experiences at home, guiding kids to master basic math concepts. This work explores Spoken Language Understanding (SLU) pipeline within a task-oriented dialogue system developed for Kid Space, with cascading Automatic Speech Recognition (ASR) and Natural Language Understanding (NLU) components evaluated on our home deployment data with kids going through gamified math learning activities. We validate the advantages of a multi-task architecture for NLU and experiment with a diverse set of pretrained language representations for Intent Recognition and Entity Extraction tasks in the math learning domain. To recognize kids' speech in realistic home environments, we investigate several ASR systems, including the commercial Google Cloud and the latest open-source Whisper solutions with varying model sizes. We evaluate the SLU pipeline by testing our best-performing NLU models on noisy ASR output to inspect the challenges of understanding children for math learning in authentic homes.