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In this paper, we suggest a two-player differential game model of transboundary pollution that accounts for time-dependent environmental absorption efficiency, which allows the biosphere to switch from a carbon sink to a source. We investigate the impact of negative externalities resulting from a transboundary pollution non-cooperative game wherein countries are dynamically involved. Based on a linear-quadratic specification for the instantaneous revenue function, we assess differences related to both transient path and steady state between cooperative solution, open-loop and Markov perfect Nash equilibria (MPNE). Regarding the methodological contribution of the paper, we suggest a particular structure of the conjectured value function to solve MPNE problems with multiplicative interaction between state variables in one state equation, so that third-order terms that arise in the Hamilton–Jacobi–Bellman equation are made negligible. Using a collocation procedure, we confirm the validity of the particular structure of the conjectured value function. The results suggest unexpected contrasts in terms of pollution control and environmental absorption efficiency management: (i) in the long run, an open-loop Nash equilibrium (OLNE) allows equivalent emissions to the social optimum but requires greater restoration efforts; (ii) although an MPNE is likely to end up with lower emissions and greater restoration efforts than an OLNE, it has a much greater chance of falling in the emergency area; (iii) the absence of cooperation and or precommitment becomes more costly as the initial absorption efficiency decreases; (iv) more heavily discounted MPNE strategies are less robust than OLNE to prevent irreversible switching of the biosphere from a carbon sink to a source.

This paper investigates how current and future generations are affected by commitment-based Nash equilibrium environmental strategies when the environmental absorption efficiency is susceptible to switch from a pollution sink to a source. We formulate a two-player differential game model of transboundary pollution that includes the environmental absorption efficiency as a state variable that can be enhanced thanks to restoration efforts. Based on a logarithmic specification for the instantaneous revenue function, we characterize the cooperative solution and the commitment-based Nash equilibrium strategy, and examine their differences in terms of steady state and transient behavior. We notably show that a commitment-based Nash equilibrium strategy makes it possible to prevent a definitive switching of the environmental absorption efficiency from a pollution sink to a source but imposes greater economic sacrifices on current generations than on future generations. In comparison, the cooperative solution imposes greater sacrifices on current generations in terms of revenues but it imposes lower environmental costs on both current and future generations than commitment-based Nash equilibrium strategy.

Model complexities of shallow (i.e., one-hidden-layer) networks representing highly varying multivariable { - 1 , 1 } -valued functions are studied in terms of variational norms tailored to dictionaries of network units. It is shown that bounds on these norms define classes of functions computable by networks with constrained numbers of hidden units and sizes of output weights. Estimates of probabilistic distributions of values of variational norms with respect to typical computational units, such as perceptrons and Gaussian kernel units, are derived via geometric characterization of variational norms combined with the probabilistic Chernoff Bound. It is shown that almost any randomly chosen { - 1 , 1 } -valued function on a sufficiently large d-dimensional domain has variation with respect to perceptrons depending on d exponentially.

Summary A technique is presented for the identification of the areas subject to flooding hazard. Starting from remote sensed elevation data and existing flood hazard maps – usually available for limited areas – the relationships between selected quantitative morphologic features and the flooding hazard are first identified and then used to extend the hazard information to the entire catchment. This is performed through techniques of pattern classification, such as linear classifiers based on quantitative morphologic features, and support vector machines with linear and Gaussian kernels. The experiment starts by discriminating between flood-prone areas and marginal hazard areas. Multiclass classifiers are subsequently used to graduate the hazard. Their designs amount to solving suitable optimization problems. Several performance measures are considered in comparing the different classifiers, such as the area under the receiver operating characteristics curve, and the sum of the false positive and false negative rates. The procedure has been validated for the Tanaro basin, a tributary to the major Italian river, the Po. Results show a high reliability: the classifier properly identifies 93% of flood-prone areas, and only 14% of the areas subject to a marginal hazard are improperly assigned. An increase of this latter value up to 19% is detected when the same structure is applied for hazard graduation. Results derived from the application to different catchments seem to qualitatively indicate the ability of the classifier to perform well also outside the calibration region. Pattern classification techniques should be considered when the identification of flood-prone areas and hazard grading is required for large regions (e.g., for civil protection or insurance purposes) or when a first identification is needed (e.g., to address further detailed flood-mapping activities).

Approximate solutions to inhomogeneous Fredholm integral equations of the second kind by radial and kernel networks are investigated. Upper bounds are derived on errors in approximation of solutions of these equations by networks with increasing model complexity. The bounds are obtained using results from nonlinear approximation theory. The results are applied to networks with Gaussian and kernel units and illustrated by numerical simulations.

A variational norm associated with sets of computational units and used in function approximation, learning from data, and infinite-dimensional optimization is investigated. For sets Gk obtained by varying a vector y of parameters in a fixed-structure computational unit K(-,y) (e.g., the set of Gaussians with free centers and widths), upper and lower bounds on the GK -variation norms of functions having certain integral representations are given, in terms of the £1-norms of the weighting functions in such representations. Families of functions for which the two norms are equal are described.

State estimation for a class of non-linear, continuous-time dynamic systems affected by disturbances is investigated. The estimator is assigned a given structure that depends on an innovation function taking on the form of a ridge computational model, with some parameters to be optimized. The behaviour of the estimation error is analysed by using input-to-state stability. The design of the estimator is reduced to the determination of the parameters in such a way as to guarantee the regional exponential stability of the estimation error in a disturbance-free setting and to minimize a cost function that measures the effectiveness of the estimation when the system is affected by disturbances. Stability is achieved by constraining the derivative of a Lyapunov function to be negative definite on a grid of points, via the penalization of the constraints that are not satisfied. Low-discrepancy sampling techniques, typical of quasi-Monte Carlo methods, are exploited in order to reduce the computational burden in ...

The approximation of the optimal policy functions is investigated for dynamic optimization problems with an objective that is additive over a finite number of stages. The distance between optimal and suboptimal values of the objective functional is estimated, in terms of the errors in approximating the optimal policy functions at the various stages. Smoothness properties are derived for such functions and exploited to choose the approximating families. The approximation error is measured in the supremum norm, in such a way to control the error propagation from stage to stage. Nonlinear approximators corresponding to Gaussian radial-basis-function networks with adjustable centers and widths are considered. Conditions are defined, guaranteeing that the number of Gaussians (hence, the number of parameters to be adjusted) does not grow “too fast” with the dimension of the state vector. The results help to mitigate the curse of dimensionality in dynamic optimization. An example of application is given and the use of the estimates is illustrated via a numerical simulation.

Complexity of Gaussian-radial-basis-function networks, with varying widths, is investigated. Upper bounds on rates of decrease of approximation errors with increasing number of hidden units are derived. Bounds are in terms of norms measuring smoothness (Bessel and Sobolev norms) multiplied by explicitly given functions a(r,d) of the number of variables d and degree of smoothness r. Estimates are proven using suitable integral representations in the form of networks with continua of hidden units computing scaled Gaussians and translated Bessel potentials. Consequences on tractability of approximation by Gaussian-radial-basis function networks are discussed.

Dynamic Programming formally solves stochastic optimization problems with an objective that is additive over a finite number of stages. However, it provides closed-form solutions only in particular cases. In general, one has to resort to approximate methodologies. In this chapter, suboptimal solutions are searched for by approximating the decision policies via linear combinations of Gaussian and sigmoidal functions containing adjustable parameters, to be optimized together with the coefficients of the combinations. These approximation schemes correspond to Gaussian radial-basis-function networks and sigmoidal feedforward neural networks, respectively. The accuracies of the suboptimal solutions are investigated by estimating the error propagation through the stages. As a case study, we address a multidimensional problem of optimal consumption under uncertainty, modeled as a stochastic optimization task with an objective that is additive over a finite number of stages. In the classical one-dimensional context, a consumer aims at maximizing over a given time horizon the discounted expected value of consumption of a good, where the expectation is taken with respect to a stochastic interest rate. The consumer has an initial wealth and at each time period earns an income, modeled as an exogenous input. We consider a multidimensional framework, in which there are \(d > 1\) consumers that aim at maximizing a social utility function. First we provide conditions that allow one to apply our estimates to such a problem; then we present a numerical analysis.

Model complexities of networks representing multivariable functions is studied in terms of variational norms tailored to types of network units. It is shown that the size of the variational norm reflects both the number of hidden units and sizes of output weights. Lower bounds on growth of variational norms with increasing input dimension d are derived for Gaussian units and perceptrons. It is proven that variation of the d-dimensional parity with respect to Gaussian Support Vector Machines grows exponentially with d and for large values of d, almost any randomly-chosen Boolean function has variation with respect to perceptrons depending on d exponentially.

Representations of multivariable Boolean functions by one and two-hidden-layer Heaviside perceptron networks are investigated. Sufficient conditions are given for representations with the numbers of network units depending on the input dimension d linearly and polynomially. Functions with such numbers depending on d exponentially or having some weights exponentially large are described in terms of properties of their communication matrices. A mathematical formalization of the concept of “highly-varying functions” is proposed. There is given an example of such function which can be represented by a network with two hidden layers with merely d units.

The role of input dimension d is studied in approximating, in various norms, target sets of d-variable functions using linear combinations of adjustable computational units. Results from the literature, which emphasize the number n of terms in the linear combination, are reformulated, and in some cases improved, with particular attention to dependence on d . For worst-case error, upper bounds are given in the factorized form ξ(d)κ(n) , where κ is nonincreasing (typically κ(n) ~ n-1/2). Target sets of functions are described for which the function ξ is a polynomial. Some important cases are highlighted where ξ decreases to zero as d → ∞. For target functions, extent (e.g., the size of domains in Rd where they are defined), scale (e.g., maximum norms of target functions), and smoothness (e.g., the order of square-integrable partial derivatives) may depend on d , and the influence of such dimension-dependent parameters on model complexity is considered. Results are applied to approximation and solution of optimization problems by neural networks with perceptron and Gaussian radial computational units.

In this letter, an analytical framework to model nodes in Intermittently Connected Networks (ICNs) is proposed. A relationship is derived in the z-domain between the discrete probability densities of their buffer state occupancies and the sizes of the arriving bulks. Under a fixed epidemic-routing-based forwarding strategy, expressions are obtained for the average buffer occupancy and its standard deviation with immediate protocol advantages.

Approximation of solutions of integral equations by networks with kernel units is investigated theoretically. There are derived upper bounds on speed of decrease of errors in approximation of solutions of Fredholm integral equations by kernel networks with increasing numbers of units. The estimates are obtained for Gaussian and degenerate kernels.

Various regularization techniques are investigated in supervised learning from data. Theoretical features of the associated optimization problems are studied, and sparse suboptimal solutions are searched for. Rates of approximate optimization are estimated for sequences of suboptimal solutions formed by linear combinations of n-tuples of computational units, and statistical learning bounds are derived. As hypothesis sets, reproducing kernel Hilbert spaces and their subsets are considered.

The tractability of neural-network approximation is investigated. The dependence of worst-case errors on the number of variables is studied. Estimates for Gaussian radial-basis-function and perceptron networks are derived.

Model complexity of feedforward neural networks is studied in terms of rates of variable-basis approximation. Sets of functions, for which the errors in approximation by neural networks with n hidden units converge to zero geometrically fast with increasing number n, are described. However, the geometric speed of convergence depends on parameters, which are specific for each function to be approximated. The results are illustrated by examples of estimates of such parameters for functions in infinite-dimensional Hilbert spaces.

A methodology to design state estimators for a class of nonlinear continuous-time dynamic systems that is based on neural networks and nonlinear programming is proposed. The estimator has the structure of a Luenberger observer with a linear gain and a parameterized (in general, nonlinear) function, whose argument is an innovation term representing the difference between the current measurement and its prediction. The problem of the estimator design consists in finding the values of the gain and of the parameters that guarantee the asymptotic stability of the estimation error. Toward this end, if a neural network is used to take on this function, the parameters (i.e., the neural weights) are chosen, together with the gain, by constraining the derivative of a quadratic Lyapunov function for the estimation error to be negative definite on a given compact set. It is proved that it is sufficient to impose the negative definiteness of such a derivative only on a suitably dense grid of sampling points. The gain is determined by solving a Lyapunov equation. The neural weights are searched for via nonlinear programming by minimizing a cost penalizing grid-point constraints that are not satisfied. Techniques based on low-discrepancy sequences are applied to deal with a small number of sampling points, and, hence, to reduce the computational burden required to optimize the parameters. Numerical results are reported and comparisons with those obtained by the extended Kalman filter are made.

Rates of approximation by networks with Gaussian RBFs with varying widths are investigated. For certain smooth functions, upper bounds are derived in terms of a Sobolev-equivalent norm. Coefficients involved are exponentially decreasing in the dimension. The estimates are proven using Bessel potentials as auxiliary approximating functions.

Covering numbers of precompact symmetric convex subsets of Hilbert spaces are investigated. Lower bounds are derived for sets containing orthogonal subsets with norms of their elements converging to zero sufficiently slowly. When these sets are convex hulls of sets with power-type covering numbers, the bounds are tight. The arguments exploit properties of generalized Hadamard matrices. The results are illustrated by examples from machine learning, neurocomputing, and nonlinear approximation.

Observers design is addressed for a class of continuous-time, nonlinear dynamic systems with Lipschitz nonlinearities. A full-order state estimator is considered that depends on an innovation function made up of two terms: a linear gain and a feedforward neural network that provides a nonlinear contribution. The gain and the weights of the neural network are chosen in such way to ensure the convergence of the estimation error. Such a goal is achieved by constraining the derivative of a Lyapunov function to be negative definite on a sampling grid of points. Under assumptions on the smoothness of the Lyapunov function and of the distribution of the sampling points, the negative definiteness of the derivative of the Lyapunov function is obtained by minimizing a cost function that penalizes the constraints that are not satisfied. Suitable sampling techniques allow to reduce the computational burden required by the network's weights optimization. Simulations results are presented to illustrate the effectiveness of the proposed method.

The stability of filters for nonlinear dynamic systems with Lipschitz nonlinearities is addressed. By considering the disturbances in the system and measurement equations as the inputs to the system representing the estimation error dynamics, the input-output stability of the filtering mapping associated with the estimation error dynamics is introduced. This is done with respect to signals belonging a Sobolev space W. The connection between W-stability and asymptotic stability (based on the existence of a suitable Lyapunov function) of the filtering mapping is investigated.

We present some comparisons between the approximation rates relevant to linear approximators and the rates relevant to neural networks, i.e., nonlinear approximators represented by sets of parametrized functions corresponding to a type of computational unit. Our analysis uses the concept of variation of a function with respect to a set. The comparison is made in terms of Kolmogorov n-width for linear spaces and a proper nonlinear n-width for the nonlinear context represented by neural networks. The results of this paper contribute to the theoretical understanding of the superiority of neural networks with respect to linear approximators in complex tasks, as is confirmed by a wide variety of applications (recognition of handwritten characters and spoken numerals, approximate solution of functional optimization problems from control theory, etc.).