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There are enormous amount of examples of Computation in nature, exemplified across multiple species in biology. One crucial aim for these computations across all life forms their ability to learn and thereby increase the chance of their…
Computational complexity is examined using the principle of increasing entropy. To consider computation as a physical process from an initial instance to the final acceptance is motivated because many natural processes have been recognized…
The question if a given partial solution to a problem can be extended reasonably occurs in many algorithmic approaches for optimization problems. For instance, when enumerating minimal dominating sets of a graph $G=(V,E)$, one usually…
The systems of complex analytic second order ordinary differential equations whose solutions close up to become rational curves (after analytic continuation) are characterized by the vanishing of an explicit differential invariant, and turn…
A coherent mathematical overview of computation and its generalisations is described. This conceptual framework is sufficient to comfortably host a wide range of contemporary thinking on embodied computation and its models.
For the ordinary differential equation (ODE) $\dot{x}(t) = f(t,x)$, $x(0) = x_0$, $t\geq 0$, $x\in R^d$, assume $f$ to be at least continuous in $t$ and locally Lipshitz in $x$, and if necessary, several times continuously differentiable in…
We overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness. The specific equations manipulated are linear elliptic, and arise in the context of…
Partial differential equations (PDEs) are widely used across the physical and computational sciences. Decades of research and engineering went into designing fast iterative solution methods. Existing solvers are general purpose, but may be…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
The infinite time-evolving block decimation (iTEBD) algorithm [Phys. Rev. Lett. 98, 070201 (2007)] allows to simulate unitary evolution and to compute the ground state of one-dimensional quantum lattice systems in the thermodynamic limit.…
Computation is commonly defined as the execution of abstract algorithms over symbolic representations, with physical systems treated as substrates that realise predefined operations. While effective for engineered machines, this separation…
Practical optimization problems may contain different kinds of difficulties that are often not tractable if one relies on a particular optimization method. Different optimization approaches offer different strengths that are good at…
We revisit evaluation of logical formulas that allow both uninterpreted relations, constrained to be finite, as well as an interpreted vocabulary over an infinite domain. This formalism was denoted embedded finite model theory in the past.…
Systems modeled by partial differential equations (PDEs) are at least as ubiquitous as systems that are by nature finite-dimensional and modeled by ordinary differential equations (ODEs). And yet, systematic and readily usable…
We consider the primal and dual forms of the optimality conditions for PDE-contrained optimization problems arising in Data-Driven Computational Mechanics when specialized to the reaction-diffusion context. Starting with the continuous…
We study the limitations and fast-forwarding of quantum algorithms for linear ordinary differential equation (ODE) systems with a particular focus on non-quantum dynamics, where the coefficient matrix in the ODE is not anti-Hermitian or the…
Fractional differential equations (FDEs) are an extension of the theory of fractional calculus. However, due to the difficulty in finding analytical solutions, there have not been extensive applications of FDEs until recent decades. With…
Finite precision computations using digital computers involve the following inherent errors: (1) Round-off error of finite precision computations (2) Binary computer arithmetic precludes exact number representation of traditional decimal…
Neural Differential Equations (NDEs) excel at modeling continuous-time dynamics, effectively handling challenges such as irregular observations, missing values, and noise. Despite their advantages, NDEs face a fundamental challenge in…
Ordinary differential equations (ODEs) are commonly used to model dynamic behavior of a system. Because many parameters are unknown and have to be estimated from the observed data, there is growing interest in statistics to develop…