Neural ODEs with stochastic vector field mixtures

It was recently shown that neural ordinary differential equation models cannot solve fundamental and seemingly straightforward tasks even with high-capacity vector field representations. This paper introduces two other fundamental tasks to the set that baseline methods cannot solve, and proposes mixtures of stochastic vector fields as a model class that is capable of solving these essential problems. Dynamic vector field selection is of critical importance for our model, and our approach is to propagate component uncertainty over the integration interval with a technique based on forward filtering. We also formalise several loss functions that encourage desirable properties on the trajectory paths, and of particular interest are those that directly encourage fewer expected function evaluations. Experimentally, we demonstrate that our model class is capable of capturing the natural dynamics of human behaviour; a notoriously volatile application area. Baseline approaches cannot adequately model this problem

The extended symplectic pencil and the finite-horizon LQ problem with two-sided boundary conditions

This note introduces a new analytic approach to the solution of a very general class of finite-horizon optimal control problems formulated for discrete-time systems. This approach provides a parametric expression for the optimal control sequences, as well as the corresponding optimal state trajectories, by exploiting a new decomposition of the so-called extended symplectic pencil. Importantly, the results established in this paper hold under assumptions that are weaker than the ones considered in the literature so far. Indeed, this approach does not require neither the regularity of the symplectic pencil, nor the modulus controllability of the underlying system. In the development of the approach presented in this paper, several ancillary results of independent interest on generalised Riccati equations and on the eigenstructure of the extended symplectic pencil will also be presented

On membrane interactions and a three-dimensional analog of Riemann surfaces

Membranes in M-theory are expected to interact via splitting and joining processes. We study these effects in the pp-wave matrix model, in which they are associated with transitions between states in sectors built on vacua with different numbers of membranes. Transition amplitudes between such states receive contributions from BPS instanton configurations interpolating between the different vacua. Various properties of the moduli space of BPS instantons are known, but there are very few known examples of explicit solutions. We present a new approach to the construction of instanton solutions interpolating between states containing arbitrary numbers of membranes, based on a continuum approximation valid for matrices of large size. The proposed scheme uses functions on a two-dimensional space to approximate matrices and it relies on the same ideas behind the matrix regularisation of membrane degrees of freedom in M-theory. We show that the BPS instanton equations have a continuum counterpart which can be mapped to the three-dimensional Laplace equation through a sequence of changes of variables. A description of configurations corresponding to membrane splitting/joining processes can be given in terms of solutions to the Laplace equation in a three-dimensional analog of a Riemann surface, consisting of multiple copies of R^3 connected via a generalisation of branch cuts. We discuss various general features of our proposal and we also present explicit analytic solutions.Comment: 64 pages, 17 figures. V2: An appendix, a figure and references added; various minor changes and improvement

Inversion of seismic reflection data from the Gialo Field, Sirte Basin

This project is concerned with the development of software to invert seismic reflection data for acoustic impedance, with application to the YY-reservoir area in Gialo Field, Sirte Basin. The problem was that of inverting post-stack seismic reflection data from two seismic lines into impedance profiles. The main input to the inversion process is an initial guess, or initial earth model, of the impedance profile defined in terms of parameters. These parameters describe the impedance and the geometry of the number of layers that constitute the earth model. Additionally, an initial guess is needed for the seismic wavelet, defined in the frequency domain using nine parameters. The inversion is an optimisation problem subject to constraints. The optimisation problem is that of minimising the error energy function defined by the sum of squares of the residuals between the observed seismic trace and its prediction by the forward model for the given earth model parameters. To determine the solution we use the method of generalised linear inverses. The generalised inverse is possible only when the Hessian matrix, which describe the curvature of error energy surface, is positive definite. When the Hessian is not definite, it is necessary to modify it to obtain the nearest positive definite matrix. To modify the Hessian we used a method based on the Cholesky factorisation. Because the modified Hessian is positive definite, we need to find the generalised inverse only once. But we may need to restrict the step-length to obtain the minimum. Such a method is a step-length based method. A step-length based method was implemented using linear equality and inequality constraints into a computer program to invert the observed seismic data for impedance. The linear equality and inequality constraints were used so that solutions that are geologically feasible and numerically stable are obtained. The strategy for the real data inversion was to first estimate the seismic wavelet at the well, then optimise the wavelet parameters. Then use the optimum wavelet to invert for impedance and layer boundaries in the seismic traces. In the three real data examples studied, this inversion scheme proved that the delineation of the Chadra sands in Gialo Field is possible. Better results could be obtained by using initial earth models that properly parameterise the subsurface, and linear constraints that are based on well data. Defining the wavelet parameters in the time domain may prove to be more stable and could lead to better inversion results