Training Neural Networks Using Reproducing Kernel Space Interpolation and Model Reduction

We introduce and study the theory of training neural networks using interpolation techniques from reproducing kernel Hilbert space theory. We generalize the method to Krein spaces, and show that widely-used neural network architectures are subsets of reproducing kernel Krein spaces (RKKS). We study the concept of "associated Hilbert spaces" of RKKS and develop techniques to improve upon the expressivity of various activation functions. Next, using concepts from the theory of functions of several complex variables, we prove a computationally applicable, multidimensional generalization of the celebrated Adamjan- Arov-Krein (AAK) theorem. The theorem yields a novel class of neural networks, called Prolongation Neural Networks (PNN). We demonstrate that, by applying the multidimensional AAK theorem to gain a PNN, one can gain performance superior to both our interpolatory methods and current state-of-the-art methods in noisy environments. We provide useful illustrations of our methods in practice

多次元システムの状態空間モデルの低次数化

秋田県立大学令和元年

Geometric Fault Detection and Isolation of Infinite Dimensional Systems

A broad class of dynamical systems from chemical processes to flexible mechanical structures, heat transfer and compression processes in gas turbine engines are represented by a set of partial differential equations (PDE). These systems are known as infinite dimensional (Inf-D) systems. Most of Inf-D systems, including PDEs and time-delayed systems can be represented by a differential equation in an appropriate Hilbert space. These Hilbert spaces are essentially Inf-D vector spaces, and therefore, they are utilized to represent Inf-D dynamical systems. Inf-D systems have been investigated by invoking two schemes, namely approximate and exact methods. Both approaches extend the control theory of ordinary differential equation (ODE) systems to Inf-D systems, however by utilizing two different methodologies. In the former approach, one needs to first approximate the original Inf-D system by an ODE system (e.g. by using finite element or finite difference methods) and then apply the established control theory of ODEs to the approximated model. On the other hand, in the exact approach, one investigates the Inf-D system without using any approximation. In other words, one first represents the system as an Inf-D system and then investigates it in the corresponding Inf-D Hilbert space by extending and generalizing the available results of finite-dimensional (Fin-D) control theory. It is well-known that one of the challenging issues in control theory is development of algorithms such that the controlled system can maintain the required performance even in presence of faults. In the literature, this property is known as fault tolerant control. The fault detection and isolation (FDI) analysis is the first step in order to achieve this goal. For Inf-D systems, the currently available results on the FDI problem are quite limited and restricted. This thesis is mainly concerned with the FDI problem of the linear Inf-D systems by using both approximate and exact approaches based on the geometric control theory of Fin-D and Inf-D systems. This thesis addresses this problem by developing a geometric FDI framework for Inf-D systems. Moreover, we implement and demonstrate a methodology for applying our results to mathematical models of a heat transfer and a two-component reaction-diffusion processes. In this thesis, we first investigate the development of an FDI scheme for discrete-time multi-dimensional (nD) systems that represent approximate models for Inf-D systems. The basic invariant subspaces including unobservable and unobservability subspaces of one-dimensional (1D) systems are extended to nD models. Sufficient conditions for solvability of the FDI problem are provided, where an LMI-based approach is also derived for the observer design. The capability of our proposed FDI methodology is demonstrated through numerical simulation results to an approximation of a hyperbolic partial differential equation system of a heat exchanger that is represented as a two-dimensional (2D) system. In the second part, an FDI methodology for the Riesz spectral (RS) system is investigated. RS systems represent a large class of parabolic and hyperbolic PDE in Inf-D systems framework. This part is mainly concerned with the equivalence of different types of invariant subspaces as defined for RS systems. Necessary and sufficient conditions for solvability of the FDI problem are developed. Moreover, for a subclass of RS systems, we first provide algorithms (for computing the invariant subspaces) that converge in a finite and known number of steps and then derive the necessary and sufficient conditions for solvability of the FDI problem. Finally, by generalizing the results that are developed for RS systems necessary and sufficient conditions for solvability of the FDI problem in a general Inf-D system are derived. Particularly, we first address invariant subspaces of Fin-D systems from a new point of view by invoking resolvent operators. This approach enables one to extend the previous Fin-D results to Inf-D systems. Particularly, necessary and sufficient conditions for equivalence of various types of conditioned and controlled invariant subspaces of Inf-D systems are obtained. Duality properties of Inf-D systems are then investigated. By introducing unobservability subspaces for Inf-D systems the FDI problem is formally formulated, and necessary and sufficient conditions for solvability of the FDI problem are provided

Structure-Preserving Model Reduction of Physical Network Systems

This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p

EVOLUTION OF THE SUBCONTINENTAL LITHOSPHERE DURING MESOZOIC TETHYAN RIFTING: CONSTRAINTS FROM THE EXTERNAL LIGURIAN MANTLE SECTION (NORTHERN APENNINE, ITALY)

Our study is focussed on mantle bodies from the External Ligurian ophiolites, within the Monte Gavi and Monte Sant'Agostino areas. Here, two distinct pyroxenite-bearing mantle sections were recognized, mainly based on their plagioclase-facies evolution. The Monte Gavi mantle section is nearly undeformed and records reactive melt infiltration under plagioclase-facies conditions. This process involved both peridotites (clinopyroxene-poor lherzolites) and enclosed spinel pyroxenite layers, and occurred at 0.7–0.8 GPa. In the Monte Gavi peridotites and pyroxenites, the spinel-facies clinopyroxene was replaced by Ca-rich plagioclase and new orthopyroxene, typically associated with secondary clinopyroxene. The reactive melt migration caused increase of TiO2 contents in relict clinopyroxene and spinel, with the latter also recording a Cr2O3 increase. In the Monte Gavi peridotites and pyroxenites, geothermometers based on slowly diffusing elements (REE and Y) record high temperature conditions (1200-1250 °C) related to the melt infiltration event, followed by subsolidus cooling until ca. 900°C. The Monte Sant'Agostino mantle section is characterized by widespread ductile shearing with no evidence of melt infiltration. The deformation recorded by the Monte Sant'Agostino peridotites (clinopyroxene-rich lherzolites) occurred at 750–800 °C and 0.3–0.6 GPa, leading to protomylonitic to ultramylonitic textures with extreme grain size reduction (10–50 μm). Compared to the peridotites, the enclosed pyroxenite layers gave higher temperature-pressure estimates for the plagioclase-facies re-equilibration (870–930 °C and 0.8–0.9 GPa). We propose that the earlier plagioclase crystallization in the pyroxenites enhanced strain localization and formation of mylonite shear zones in the entire mantle section. We subdivide the subcontinental mantle section from the External Ligurian ophiolites into three distinct domains, developed in response to the rifting evolution that ultimately formed a Middle Jurassic ocean-continent transition: (1) a spinel tectonite domain, characterized by subsolidus static formation of plagioclase, i.e. the Suvero mantle section (Hidas et al., 2020), (2) a plagioclase mylonite domain experiencing melt-absent deformation and (3) a nearly undeformed domain that underwent reactive melt infiltration under plagioclase-facies conditions, exemplified by the the Monte Sant'Agostino and the Monte Gavi mantle sections, respectively. We relate mantle domains (1) and (2) to a rifting-driven uplift in the late Triassic accommodated by large-scale shear zones consisting of anhydrous plagioclase mylonites. Hidas K., Borghini G., Tommasi A., Zanetti A. &amp; Rampone E. 2021. Interplay between melt infiltration and deformation in the deep lithospheric mantle (External Liguride ophiolite, North Italy). Lithos 380-381, 105855

Impact of geogenic degassing on C-isotopic composition of dissolved carbon in karst systems of Greece

The Earth C-cycle is complex, where endogenic and exogenic sources are interconnected, operating in a multiple spatial and temporal scale (Lee et al., 2019). Non-volcanic CO2 degassing from active tectonic structures is one of the less defined components of this cycle (Frondini et al., 2019). Carbon mass-balance (Chiodini et al., 2000) is a useful tool to quantify the geogenic carbon output from regional karst hydrosystems. This approach has been demonstrated for central Italy and may be valid also for Greece, due to the similar geodynamic settings. Deep degassing in Greece has been ascertained mainly at hydrothermal and volcanic areas, but the impact of geogenic CO2 released by active tectonic areas has not yet been quantified. The main aim of this research is to investigate the possible deep degassing through the big karst aquifers of Greece. Since 2016, 156 karst springs were sampled along most of the Greek territory. To discriminate the sources of carbon, the analysis of the isotopic composition of carbon was carried out. δ13CTDIC values vary from -16.61 to -0.91‰ and can be subdivided into two groups characterized by (a) low δ13CTDIC, and (b) intermediate to high δ13CTDIC with a threshold value of -6.55‰. The composition of the first group can be related to the mixing of organic-derived CO2 and the dissolution of marine carbonates. Springs of the second group, mostly located close to Quaternary volcanic areas, are linked to possible carbon input from deep sources