D-MORPH regression: application to modeling with unknown parameters more than observation data

Diffeomorphic modulation under observable response preserving homotopy (D-MORPH) is a model exploration method, originally developed for differential equations. We extend D-MORPH to regression treatment of a model described as a linear superposition of basis functions with unknown parameters being the expansion coefficients. The goal of D-MORPH regression is to improve prediction accuracy without sacrificing fitting accuracy. When there are more unknown parameters than observation data, the corresponding linear algebraic equation system is generally consistent, and has an infinite number of solutions exactly fitting the data. In this case, the solutions given by standard regression techniques can significantly deviate from the true system structure, and consequently provide large prediction errors for the model. D-MORPH regression is a practical systematic means to search over system structure within the infinite number of possible solutions while preserving fitting accuracy. An explicit expression is provided by D-MORPH regression relating the data to the expansion coefficients in the linear model. The expansion coefficients obtained by D-MORPH regression are particular linear combinations of those obtained by least-squares regression. The resultant prediction accuracy provided by D-MORPH regression is shown to be significantly improved in several model illustrations.     

Cooperating or fighting with decoherence in the optimal control of quantum dynamics

This paper explores the use of laboratory closed-loop learning control to either fight or cooperate with decoherence in the optimal manipulation of quantum dynamics. Simulations of the processes are performed in a Lindblad formulation on multilevel quantum systems strongly interacting with the environment without spontaneous emission. When seeking a high control yield it is possible to find fields that successfully fight with decoherence while attaining a good quality yield. When seeking modest control yields, fields can be found which are optimally shaped to cooperate with decoherence and thereby drive the dynamics more efficiently. In the latter regime when the control field and the decoherence strength are both weak, a theoretical foundation is established to describe how they cooperate with each other. In general, the results indicate that the population transfer objectives can be effectively met by appropriately either fighting or cooperating with decoherence.     

High Dimensional Model Representations Generated from Low Dimensional Data Samples. I. mp-Cut-HDMR

High dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for improving the efficiency of deducing high dimensional input–output system behavior. For a high dimensional system, an output f(x) is commonly a function of many input variables x=|x ₁,x ₂,...,x _n } with n10² or larger. HDMR describes f(x) by a finite hierarchical correlated function expansion in terms of the input variables. Various forms of HDMR can be constructed for different purposes. Cut- and RS-HDMR are two particular HDMR expansions. Since the correlated functions in an HDMR expansion are optimal choices tailored to f(x) over the entire domain of x, the high order terms (usually larger than second order, or beyond pair cooperativity) in the expansion are often negligible. When the approximations given by the first and the second order Cut-HDMR correlated functions are not adequate, this paper presents a monomial based preconditioned HDMR method to represent the higher order terms of a Cut-HDMR expansion by expressions similar to the lower order ones with monomial multipliers. The accuracy of the Cut-HDMR expansion can be significantly improved using preconditioning with a minimal number of additional input–output samples without directly invoking the determination of higher order terms. The mathematical foundations of monomial based preconditioned Cut-HDMR is presented along with an illustration of its applicability to an atmospheric chemical kinetics model.     

Regularized random-sampling high dimensional model representation (RS-HDMR)

High Dimensional Model Representation (HDMR) is under active development as a set of quantitative model assessment and analysis tools for capturing high-dimensional input–output system behavior. HDMR is based on a hierarchy of component functions of increasing dimensions. The Random-Sampling High Dimensional Model Representation (RS-HDMR) is a practical approach to HDMR utilizing random sampling of the input variables. To reduce the sampling effort, the RS-HDMR component functions are approximated in terms of a suitable set of basis functions, for instance, orthonormal polynomials. Oscillation of the outcome from the resultant orthonormal polynomial expansion can occur producing interpolation error, especially on the input domain boundary, when the sample size is not large. To reduce this error, a regularization method is introduced. After regularization, the resultant RS-HDMR component functions are smoother and have better prediction accuracy, especially for small sample sizes (e.g., often few hundred). The ignition time of a homogeneous H₂/air combustion system within the range of initial temperature, 1000 < T ₀ < 1500 K, pressure, 0.1 < P < 100 atm and equivalence ratio of H₂/O₂, 0.2 < R < 10 is used for testing the regularized RS-HDMR.      

D-MORPH regression for modeling with fewer unknown parameters than observation data

D-MORPH regression is a procedure for the treatment of a model prescribed as a linear superposition of basis functions with less observation data than the number of expansion parameters. In this case, there is an infinite number of solutions exactly fitting the data. D-MORPH regression provides a practical systematic means to search over the solutions seeking one with desired ancillary properties while preserving fitting accuracy. This paper extends D-MORPH regression to consider the common case where there is more observation data than unknown parameters. This situation is treated by utilizing a proper subset of the normal equation of least-squares regression to judiciously reduce the number of linear algebraic equations to be less than the number of unknown parameters, thereby permitting application of D-MORPH regression. As a result, no restrictions are placed on model complexity, and the model with the best prediction accuracy can be automatically and efficiently identified. Ignition data for a H ₂/air combustion model as well as laboratory data for quantum-control-mechanism identification are used to illustrate the method.     

Resolution of strongly competitive product channels with optimal dynamic discrimination: application to flavins

Fundamental molecular selectivity limits are probed by exploiting laser-controlled quantum interferences for the creation of distinct spectral signatures in two flavin molecules, erstwhile nearly indistinguishable via steady-state methods. Optimal dynamic discrimination (ODD) uses optimally shaped laser fields to transiently amplify minute molecular variations that would otherwise go unnoticed with linear absorption and fluorescence techniques. ODD is experimentally demonstrated by combining an optimally shaped UV pump pulse with a time-delayed, fluorescence-depleting IR pulse for discrimination amongst riboflavin and flavin mononucleotide in aqueous solution, which are structurally and spectroscopically very similar. Closed-loop, adaptive pulse shaping discovers a set of UV pulses that induce disparate responses from the two flavins and allows for concomitant flavin discrimination of ～16σ. Additionally, attainment of ODD permits quantitative, analytical detection of the individual constituents in a flavin mixture. The successful implementation of ODD on quantum systems of such high complexity bodes well for the future development of the field and the use of ODD techniques in a variety of demanding practical applications.     

Parametric scaling of mathematical models

Parametric scaling, the process of extrapolation of a modelling result to new parametric conditions, is often required in model optimization, and can be important if the effects of parametric uncertainty on model predictions are to be quantified. Knowledge of the functional relationship between the model solution (y) and the system parameters (α) may also provide insight into the physical system underlying the model. This paper examines strategies for parametric scaling, assuming that only the nominal model solution y(α) and the associated parametric sensitivity coefficients (?y/?α, ?²y/?α², etc.) are known. The truncated Taylor series is shown to be a poor choice for parametric scaling, when y has known bounds. Alternate formulae are proposed which ‘build-in’ the constraints on y, thus expanding the parametric region in which the extrapolation may be valid. In the case where y has a temporal as well as a parametric dependence, the extrapolation may be further improved by removing from the Taylor series coefficients the ‘secular’ components, which refer to changes in the time scale of y(t), not to changes in y as a function of α.     

Efficient chemical kinetic modeling through neural network maps

An approach to modeling nonlinear chemical kinetics using neural networks is introduced. It is found that neural networks based on a simple multivariate polynomial architecture are useful in approximating a wide variety of chemical kinetic systems. The accuracy and efficiency of these ridge polynomial networks (RPNs) are demonstrated by modeling the kinetics of H(2) bromination, formaldehyde oxidation, and H(2)+O(2) combustion. RPN kinetic modeling has a broad range of applications, including kinetic parameter inversion, simulation of reactor dynamics, and atmospheric modeling.     

A scalable algorithm for molecular property estimation in high dimensional scaffold-based libraries

An algorithm is presented for the estimation of molecular properties over a library built around a scaffold, which has N sites for functionalization with M _i moieties at the ith scaffold site, corresponding to a library of ${\prod_{i=1}^N M_i}$ molecules. The algorithm relies on a series of operations involving (i) synthesis and property measurement of a minimal number of T randomly sampled members of the library, (ii) expression of the observed property in terms of a high-dimensional model representation (HDMR) of the moiety → property map, (iii) optimization of the ordered sequence of moieties on each site to regularize the HDMR map and (iv) interpolation using the map to estimate the properties of as yet unsynthesized compounds. The set of operations is performed iteratively aiming to reach convergence of the predictive HDMR map with as few synthesized samples as possible. Through simulation, the number T of required random molecular samples is shown to scale very favorably with ${T < < \prod^N_{i=1} M_i}$ for cases up to N = 20 and M _i = 20. For example, high estimation quality was attained for simulated libraries with T ~ 5,000 sampled compounds for a library of 20¹² members and T ~ 12,500 sampled compounds for a library of 20²⁰ members. The algorithm is based on the assumption that a systematic pattern exists in the moiety → property map provided that the moieties are optimally ordered on the scaffold sites within the context of HDMR. The overall procedure is referred to as the substituent reordering HDMR algorithm (SR-HDMR). The technique was also successfully tested with laboratory data for estimating C¹³-NMR shifts in a tri-substituted benzene library and for lac operon repression binding.