Related papers: The Glider Equation for Asymptotic Lenia
Like Life, Lenia CA support a range of patterns that move, interact with their environment, and/or are modified by said interactions. These patterns maintain a cohesive, self-organizing morphology, i.e. they exemplify autopoiesis, the…
We report in experiment and simulation the spontaneous formation of dynamically bound pairs of shape changing smarticle robots undergoing locally repulsive collisions. Borrowing terminology from Conway's simulated Game of Life, these…
Substantial efforts have been applied to engineer CA with desired emergent properties, such as supporting gliders. Recent work in continuous CA has generated a wide variety of compelling bioreminiscent patterns, and the expansion of CA…
Many simulated complex systems that support persistent self-organizing patterns, i.e. gliders, have a 'state-plus-update' paradigm. This approach can be found in computational models of physics, continuous and neural cellular automata,…
Lenia is a continuous generalization of Conway's Game of Life. Bert Wang-Chak Chan has discovered and published many seemingly organic dynamics in his Lenia simulations since 2019. These simulations follow the Euler curve algorithm starting…
We asymptotically derive a non-linear Langevin-like equation with non-Gaussian white noise for a wide class of stochastic systems associated with multiple stochastic environments, by developing the expansion method in our previous paper [K.…
Learning rules -- prescriptions for updating model parameters to improve performance -- are typically assumed rather than derived. Why do some learning rules work better than others, and under what assumptions can a given rule be considered…
Natural gradient descent is a principled method for adapting the parameters of a statistical model on-line using an underlying Riemannian parameter space to redefine the direction of steepest descent. The algorithm is examined via methods…
We are concerned with the dynamical behavior of solutions to semilinear wave systems with time-varying damping and nonconvex force potential. Our result shows that the dynamical behavior of solution is asymptotically stable without any…
The subject of the article is linear systems of wave equations on cosmological backgrounds with convergent asymptotics. The condition of convergence corresponds to the requirement that the second fundamental form, when suitably normalised,…
Without loss of generalisation to other systems, including possibly non-deterministic ones, we demonstrate the application of methods drawn from algorithmic information dynamics to the characterisation and classification of emergent and…
We examine a class of stochastic differential inclusions involving multiscale effects designed to solve a class of generalized variational inequalities. This class of problems contains constrained convex non-smooth optimization problems,…
In this paper we consider the global stability of solutions of a nonlinear stochastic differential equation. The differential equation is a perturbed version of a globally stable linear autonomous equation with unique zero equilibrium where…
The dynamical stability of the iterates during training plays a key role in determining the minima obtained by optimization algorithms. For example, stable solutions of gradient descent (GD) correspond to flat minima, which have been…
Continuous cellular automata (CCAs) have evolved from discrete lookup tables to continuous partial differential equation (PDE) formulations in the search for novel forms of complexity. Despite innovations in qualitative behavior, analytical…
In this paper, we study the asymptotic behavior of solutions to a scalar fractional delay differential equations around the equilibrium points. More precise, we provide conditions on the coefficients under which a linear fractional delay…
We develop a new approach to drifting games, a class of two-person games with many applications to boosting and online learning settings. Our approach involves (a) guessing an asymptotically optimal potential by solving an associated…
We analyse the underlying nonlinear partial differential equation which arises in the study of gravitating flat fluid plates of embedding class one. Our interest in this equation lies in discussing new solutions that can be found by means…
In this paper we characterise the global stability, global boundedness and recurrence of solutions of a scalar nonlinear stochastic differential equation. The differential equation is a perturbed version of a globally stable autonomous…
In this paper we consider the global stability of solutions of an affine stochastic differential equation. The differential equation is a perturbed version of a globally stable linear autonomous equation with unique zero equilibrium where…