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Problems that are not well-posed in the sense above are termed ill-posed. A simple example is a global optimization problem, because the location of the optima is generally not a continuous function of the parameters specifying the objective, even when the objective itself is a smooth function of those parameters.
Manifold regularization is a type of regularization, a family of techniques that reduces overfitting and ensures that a problem is well-posed by penalizing complex solutions. In particular, manifold regularization extends the technique of Tikhonov regularization as applied to Reproducing kernel Hilbert spaces (RKHSs).
This was first introduced by Tikhonov [4] to solve ill-posed problems. Many statistical learning algorithms can be expressed in such a form (for example, the well-known ridge regression ). The tradeoff between ( a ) {\displaystyle (a)} and ( b ) {\displaystyle (b)} in ( 2 ) is geometrically more intuitive with Tikhonov regularization in RKHS.
In mathematics, statistics, finance, [1] and computer science, particularly in machine learning and inverse problems, regularization is a process that converts the answer of a problem to a simpler one. It is often used in solving ill-posed problems or to prevent overfitting. [2]
Hence, the problem is ill-conditioned, and solving this RLS problem amounts to stabilizing a possibly ill-conditioned matrix inversion problem, which is studied in the theory of ill-posed inverse problems; in both problems, a main concern is to deal with the issue of numerical stability.
The analysis of these methods proceeds in two steps. First, we will show that the Galerkin equation is a well-posed problem in the sense of Hadamard and therefore admits a unique solution. In the second step, we study the quality of approximation of the Galerkin solution .
General Problem Solver (GPS) is a computer program created in 1957 by Herbert A. Simon, J. C. Shaw, and Allen Newell (RAND Corporation) intended to work as a universal problem solver machine. In contrast to the former Logic Theorist project, the GPS works with means–ends analysis .
Physics-informed neural networks for solving Navier–Stokes equations. Physics-informed neural networks (PINNs), [1] also referred to as Theory-Trained Neural Networks (TTNs), [2] are a type of universal function approximators that can embed the knowledge of any physical laws that govern a given data-set in the learning process, and can be described by partial differential equations (PDEs).