Related papers: Differential geometry and stochastic dynamics with…
To model deformation of anatomical shapes, non-linear statistics are required to take into account the non-linear structure of the data space. Computer implementations of non-linear statistics and differential geometry algorithms often lead…
When neural networks are used to solve differential equations, they usually produce solutions in the form of black-box functions that are not directly mathematically interpretable. We introduce a method for generating symbolic expressions…
DifferentialGeometry is a Maple software package which symbolically performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the…
Theano is a Python library that allows to define, optimize, and evaluate mathematical expressions involving multi-dimensional arrays efficiently. Since its introduction, it has been one of the most used CPU and GPU mathematical compilers -…
Modelers use automatic differentiation (AD) of computation graphs to implement complex Deep Learning models without defining gradient computations. Stochastic AD extends AD to stochastic computation graphs with sampling steps, which arise…
Machine learning has been progressively generalised to operate within non-Euclidean domains, but geometrically accurate methods for learning on surfaces are still falling behind. The lack of closed-form Riemannian operators, the…
Identifying governing equations for a dynamical system is a topic of critical interest across an array of disciplines, from mathematics to engineering to biology. Machine learning -- specifically deep learning -- techniques have shown their…
Automated scientific discovery aims to improve scientific understanding through machine learning. A central approach in this field is symbolic regression, which uses genetic programming or sparse regression to learn interpretable…
Symbolic encoding has been used in multi-operator learning as a way to embed additional information for distinct time-series data. For spatiotemporal systems described by time-dependent partial differential equations, the equation itself…
Executing complex manipulation in cluttered environments requires satisfying coupled geometric and temporal constraints. Although Spatio-Temporal Logic (SpaTiaL) offers a principled specification framework, its use in gradient-based…
Optimization on manifolds is a class of methods for optimization of an objective function, subject to constraints which are smooth, in the sense that the set of points which satisfy the constraints admits the structure of a differentiable…
Derivatives play a critical role in computational statistics, examples being Bayesian inference using Hamiltonian Monte Carlo sampling and the training of neural networks. Automatic differentiation is a powerful tool to automate the…
This primer explains how continuous-time stochastic processes (precisely, Brownian motion and other Ito diffusions) can be defined and studied on manifolds. No knowledge is assumed of either differential geometry or continuous-time…
We present a family of Python modules for the numerical integration of ordinary, delay, or stochastic differential equations. The key features are that the user enters the derivative symbolically and it is just-in-time-compiled, allowing…
We describe a neural-based method for generating exact or approximate solutions to differential equations in the form of mathematical expressions. Unlike other neural methods, our system returns symbolic expressions that can be interpreted…
In the domain of geometry and topology optimization, discovering geometries that optimally satisfy specific problem criteria is a complex challenge in both engineering and scientific research. In this work, we propose a new approach for the…
We propose a coordinate-invariant geometric formulation of the GENERIC stochastic differential equation, unifying reversible Hamiltonian and irreversible dissipative dynamics within a differential-geometric framework. Our construction…
Scientific machine learning has enabled the extraction of physical insights and data-driven modeling of high-dimensional spatiotemporal data, yet achieving physically interpretable latent representations and computationally efficient…
We introduce the C++ library Wedge, based on GiNaC, for symbolic computations in differential geometry. We show how Wedge makes it possible to use the language C++ to perform such computations, and illustrate some advantages of this…
We propose a probabilistic framework for developing computational models of biological neural systems. In this framework, physiological recordings are viewed as discrete-time partial observations of an underlying continuous-time stochastic…