Solution of Inverse Problems in Laser Spectroscopy of Water Media with the Help of Neural Networks

Two methodological approaches to inverse problems solution with the help of neural networks are considered: "experiment-based" and "model-based". Their merits, drawbacks, and characteristics of their use are discussed. Successful application of neural networks for solution of three inverse problems in laser spectroscopy of water media is reported: (1) simultaneous determination of sea water temperature and salinity from Raman spectra, (2) determination of contributions for components of an organic compounds mixture in water from their fluorescence spectra, and (3) determination of molecular parameters of organic compounds from fluorescence saturation curves. Inverse Problems and Their Correctness One of the areas of successful application of neural networks (NN) in the last years was the solution of various inverse problems in science and technology. The general statement of an inverse problem is the following. Consider the studied object of an arbitrary nature, whose behavior is determined by the vector of input parameters X = {x1, x2, ..., xn}. Let the value of X be unknown and the behavior of the object be expressed as a vector of observed values Y = {y1, y2, ..., ym}. Therefore, the studied object implements an unknown function Y = F(X). As a rule, m >> n, i.e. the system can in fact be described by much less parameters than observed. To do this, it is necessary to study how to restore X values from Y values. Such problem is a problem of modeling the inverse function X=F(Y) and it is called an inverse problem. Such a problem may not be always solved unambiguously because of possible non-uniqueness and instability of the solutions. An inverse problem is called correct by Hadamar, if: (1) ∀Y ∃ !X: Y=F(X) (the solution is unique); (2) ∀δ ∃ε: (∆Y < δ) ⇔ (∆X < ε) (the solution is stable) Copyright © 2001, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. in the whole domain of definition of X. An inverse problem is called correct by Tikhonov (Tikhonov, Dmitriev, and Glasko 1983) (in the case when correctness by Hadamar is breached), if it is possible to extract such more narrow set of solutions {X'} from the space {X}, that: (1) it is known a priori that ∃ X ∈ {X'} (the solution exists); (2) ∀Y ∃ !X ∈ {X'}: Y=F(X) (the solution is unique); (3) ∀δ ∃ε: (∆Y < δ) ⇔ (∆X < ε), if (X+∆X) ∈ {X'} (the solution is stable). It should be noted that even when the problem is theoretically correct, it can be practically incorrect because of presence of noise in experimental data (that may lead to instability of the solution) and because of discreteness of the set of experimental points (that may break the uniqueness of the solution – the observed values may be described by several different functions Y=F1(X), Y=F2(X) etc.). Due to the well-known properties of NN (ability to generalize the available information if the data are contradictive, and thus to rise the effective signal to noise ratio noticeably), it is possible to use them to oppose the emergence of practical incorrectness during solution of inverse problems. Two Methodological Approaches to the Solution of Inverse Problems with the Help of Neural Networks Practical solution of inverse problems with the help of NN is possible based on two principally different methodological approaches. From: FLAIRS-01 Proceedings. Copyright © 2001, AAAI (www.aaai.org). All rights reserved.