Related papers: Nilpotent Feed Forward Network Dynamics
The dynamical systems of the form $\ddot\bold r=\bold F (\bold r,\dot\bold r)$ in $\Bbb R^n$ accepting the normal shift are considered. The concept of weak normality for them is introduced. The partial differential equations for the force…
Through the Hasimoto map, various dynamical systems can be mapped to different integrodifferential generalizations of Nonlinear Schrodinger (NLS) family of equations some of which are known to be integrable. Two such continuum limits,…
Several authors have reported that the echo state network reproduces bifurcation diagrams of some nonlinear differential equations using the data for a few control parameters. We demonstrate that a simpler feedforward neural network can…
It has been established that a positive semi-definite Hamiltonian,$H$, that has a tridiagonal matrix representation in a basis set, allows a definition of forward (and backward) shift operators that can be used to define the matrix…
We propose an effective and lightweight learning algorithm, Symplectic Taylor Neural Networks (Taylor-nets), to conduct continuous, long-term predictions of a complex Hamiltonian dynamic system based on sparse, short-term observations. At…
Neural networks can implement arbitrary functions. But, mechanistically, what are the tools at their disposal to construct the target? For classification tasks, the network must transform the data classes into a linearly separable…
We analyze the dynamics of pattern forming fronts which propagate into an unstable state, and whose dynamics is of the pulled type, so that their asymptotic speed is equal to the linear spreading speed v^*. We discuss a method that allows…
In this study, we introduce a method for learning group (known or unknown) equivariant functions by learning the associated quadratic form $x^T A x$ corresponding to the group from the data. Certain groups, known as orthogonal groups,…
We find the complete equivalence group of a class of (1+1)-dimensional second-order evolution equations, which is infinite-dimensional. The equivariant moving frame methodology is invoked to construct, in the regular case of the…
We propose an abstract framework describing energy-renormalized Hamiltonians in terms of local algebras. Within the framework, we examine the positivity improvingness of the semigroup generated by the renormalized Hamiltonian. As examples,…
We develop a general mathematical framework to analyze scaling regimes and derive explicit analytic solutions for gradient flow (GF) in large learning problems. Our key innovation is a formal power series expansion of the loss evolution,…
Via a non degenerate symmetric bilinear form we identify the coadjoint representation with a new representation and so we induce on the orbits a simplectic form. By considering Hamiltonian systems on the orbits we study some features of…
In this paper, we introduce a novel class of neural differential equation, which are intrinsically Lyapunov stable, exponentially stable or passive. We take a recently proposed Polyak Lojasiewicz network (PLNet) as an Lyapunov function and…
We consider fully connected feed-forward deep neural networks (NNs) where weights and biases are independent and identically distributed as symmetric centered stable distributions. Then, we show that the infinite wide limit of the NN, under…
We study logarithmic conformal field theories (LCFTs) through the introduction of nilpotent conformal weights. Using this device, we derive the properties of LCFT's such as the transformation laws, singular vectors and the structure of…
Matrices over the ring of formal power series are considered. Normal forms with respect to various sub-groups of the two-sided transformations are constructed. The construction is based on the special property of the action: it induces a…
The absence of intrinsic adjacency relations and orientation systems in hypergraphs creates fundamental challenges for constructing sheaf Laplacians of arbitrary degrees. We resolve these limitations through symmetric simplicial sets…
Diffusion-driven instability is a fundamental mechanism underlying pattern formation in spatially extended systems. In almost all existing works, diffusion across the links of the underlying network is modeled through scalar weights,…
In this paper, we propose a new method for motion prediction: HSFM-$\Sigma$nn. Our proposed method combines two different approaches: a feedforward network whose layers are model-based transition functions using the HSFM and a Neural…
Predicting the evolution of systems that exhibit spatio-temporal dynamics in response to external stimuli is a key enabling technology fostering scientific innovation. Traditional equations-based approaches leverage first principles to…