We present a novel geometric perspective on the latent space of diffusion models. We first show that the standard pullback approach, utilizing the deterministic probability flow ODE decoder, is fundamentally flawed. It provably forces geodesics to decode as straight segments in data space, effectively ignoring any intrinsic data geometry beyond the ambient Euclidean space. Complementing this view, diffusion also admits a stochastic decoder via the reverse SDE, which enables an information geometric treatment with the Fisher-Rao metric. However, a choice of $x_T$ as the latent representation collapses this metric due to memorylessness. We address this by introducing a latent spacetime $z=(x_t,t)$ that indexes the family of denoising distributions $p(x_0 | x_t)$ across all noise scales, yielding a nontrivial geometric structure. We prove these distributions form an exponential family and derive simulation-free estimators for curve lengths, enabling efficient geodesic computation. The resulting structure induces a principled Diffusion Edit Distance, where geodesics trace minimal sequences of noise and denoise edits between data. We also demonstrate benefits for transition path sampling in molecular systems, including constrained variants such as low-variance transitions and region avoidance. Code is available at: https://github.com/rafalkarczewski/spacetime-geometry

Galilean symmetry is the natural symmetry of inertial motion that underpins Newtonian physics. Although rigid-body symmetry is one of the most established and fundamental tools in robotics, there appears to be no comparable treatment of Galilean symmetry for a robotics audience. In this paper, we present a robotics-tailored exposition of Galilean symmetry that leverages the community's familiarity with and understanding of rigid-body transformations and pose representations. Our approach contrasts with common treatments in the physics literature that introduce Galilean symmetry as a stepping stone to Einstein's relativity. A key insight is that the Galilean matrix Lie group can be used to describe two different pose representations, Galilean frames, that use inertial velocity in the state definition, and extended poses, that use coordinate velocity. We provide three examples where applying the Galilean matrix Lie-group algebra to robotics problems is straightforward and yields significant insights: inertial navigation above the rotating Earth, manipulator kinematics, and sensor data fusion under temporal uncertainty. We believe that the time is right for the robotics community to benefit from rediscovering and extending this classical material and applying it to modern problems.

We develop a flexible framework based on physics-informed neural networks (PINNs) for solving boundary value problems involving minimal surfaces in curved spacetimes, with a particular emphasis on singularities and moving boundaries. By encoding the underlying physical laws into the loss function and designing network architectures that incorporate the singular behavior and dynamic boundaries, our approach enables robust and accurate solutions to both ordinary and partial differential equations with complex boundary conditions. We demonstrate the versatility of this framework through applications to minimal surface problems in anti-de Sitter (AdS) spacetime, including examples relevant to the AdS/CFT correspondence (e.g. Wilson loops and gluon scattering amplitudes) popularly used in the context of string theory in theoretical physics. Our methods efficiently handle singularities at boundaries, and also support both "soft" (loss-based) and "hard" (formulation-based) imposition of boundary conditions, including cases where the position of a boundary is promoted to a trainable parameter. The techniques developed here are not limited to high-energy theoretical physics but are broadly applicable to boundary value problems encountered in mathematics, engineering, and the natural sciences, wherever singularities and moving boundaries play a critical role.

We present GravLensX, an innovative method for rendering black holes with gravitational lensing effects using neural networks. The methodology involves training neural networks to fit the spacetime around black holes and then employing these trained models to generate the path of light rays affected by gravitational lensing. This enables efficient and scalable simulations of black holes with optically thin accretion disks, significantly decreasing the time required for rendering compared to traditional methods. We validate our approach through extensive rendering of multiple black hole systems with superposed Kerr metric, demonstrating its capability to produce accurate visualizations with significantly $15\times$ reduced computational time. Our findings suggest that neural networks offer a promising alternative for rendering complex astrophysical phenomena, potentially paving a new path to astronomical visualization.

We introduce Einstein Fields, a neural representation that is designed to compress computationally intensive four-dimensional numerical relativity simulations into compact implicit neural network weights. By modeling the \emph{metric}, which is the core tensor field of general relativity, Einstein Fields enable the derivation of physical quantities via automatic differentiation. However, unlike conventional neural fields (e.g., signed distance, occupancy, or radiance fields), Einstein Fields are \emph{Neural Tensor Fields} with the key difference that when encoding the spacetime geometry of general relativity into neural field representations, dynamics emerge naturally as a byproduct. Einstein Fields show remarkable potential, including continuum modeling of 4D spacetime, mesh-agnosticity, storage efficiency, derivative accuracy, and ease of use. We address these challenges across several canonical test beds of general relativity and release an open source JAX-based library, paving the way for more scalable and expressive approaches to numerical relativity. Code is made available at https://github.com/AndreiB137/EinFields

We show that there is a fast algorithm that embeds hierarchical structures in three-dimensional Minkowski spacetime. The correlation of data ends up purely encoded in the causal structure. Our model relies solely on oriented token pairs -- local hierarchical signals -- with no access to global symbolic structure. We apply our method to the corpus of \textit{WordNet}. We provide a perfect embedding of the mammal sub-tree including ambiguities (more than one hierarchy per node) in such a way that the hierarchical structures get completely codified in the geometry and exactly reproduce the ground-truth. We extend this to a perfect embedding of the maximal unambiguous subset of the \textit{WordNet} with 82{,}115 noun tokens and a single hierarchy per token. We introduce a novel retrieval mechanism in which causality, not distance, governs hierarchical access. Our results seem to indicate that all discrete data has a perfect geometrical representation that is three-dimensional. The resulting embeddings are nearly conformally invariant, indicating deep connections with general relativity and field theory. These results suggest that concepts, categories, and their interrelations, namely hierarchical meaning itself, is geometric.

From H. G. Wells's The Time Machine to Christopher Nolan's Interstellar, the possibility of travelling through time has fascinated people for centuries. But, although it sounds like pure science fiction, physicists now believe that time travel really is possible. In fact, scientists say that people have already done it. But, before you start to plan your trip to ancient Rome, the experts caution that real time travel is nothing like what you see in the movies. It might seem obvious, but here on Earth, we all move through time at a speed of one second per second.

Field-based reactive control provides a minimalist, decentralized route to guiding robots that lack onboard computation. Such schemes are well suited to resource-limited machines like microrobots, yet implementation artifacts, limited behaviors, and the frequent lack of formal guarantees blunt adoption. Here, we address these challenges with a new geometric approach called artificial spacetimes. We show that reactive robots navigating control fields obey the same dynamics as light rays in general relativity. This surprising connection allows us to adopt techniques from relativity and optics for constructing and analyzing control fields. When implemented, artificial spacetimes guide robots around structured environments, simultaneously avoiding boundaries and executing tasks like rallying or sorting, even when the field itself is static. We augment these capabilities with formal tools for analyzing what robots will do and provide experimental validation with silicon-based microrobots. Combined, this work provides a new framework for generating composed robot behaviors with minimal overhead.

We introduce a novel interpretable Neural Network (NN) model designed to perform precision bulk reconstruction under the AdS/CFT correspondence. According to the correspondence, a specific condensed matter system on a ring is holographically equivalent to a gravitational system on a bulk disk, through which tabletop quantum gravity experiments may be possible as reported in arXiv:2211.13863. The purpose of this paper is to reconstruct a higher-dimensional gravity metric from the condensed matter system data via machine learning using the NN. Our machine reads spatially and temporarily inhomogeneous linear response data of the condensed matter system, and incorporates a novel layer that implements the Runge-Kutta method to achieve better numerical control. We confirm that our machine can let a higher-dimensional gravity metric be automatically emergent as its interpretable weights, using a linear response of the condensed matter system as data, through supervised machine learning. The developed method could serve as a foundation for generic bulk reconstruction, i.e., a practical solution to the AdS/CFT correspondence, and would be implemented in future tabletop quantum gravity experiments.

We present an explicit construction of a relativistic quantum computing architecture using a variational quantum circuit approach that is shown to allow for universal quantum computing. The variational quantum circuit consists of tunable single-qubit rotations and entangling gates that are implemented successively. The single qubit rotations are parameterized by the proper time intervals of the qubits' trajectories and can be tuned by varying their relativistic motion in spacetime. The entangling layer is mediated by a relativistic quantum field instead of through direct coupling between the qubits. Within this setting, we give a prescription for how to use quantum field-mediated entanglement and manipulation of the relativistic motion of qubits to obtain a universal gate set, for which compact non-perturbative expressions that are valid for general spacetimes are also obtained. We also derive a lower bound on the channel fidelity that shows the existence of parameter regimes in which all entangling operations are effectively unitary, despite the noise generated from the presence of a mediating quantum field. Finally, we consider an explicit implementation of the quantum Fourier transform with relativistic qubits.