Rigorous Theorems Research Articles

The Necessity of Machine Learning Theory in Mitigating AI Risk

Highlights SUMMARY In the last years we have witnessed rapidly accelerating progress in Neural Network-based Artificial Intelligence. Yet our fundamental understanding of these methods has lagged far behind. Never before had a technology been developed so quickly and deployed so widely with so little understanding of its basic principles. In this document we argue that deep learning systems with their human-like complexity cannot be controlled and guided in socially acceptable ways or countered in adversarial situations, without an underlying theory. Theory in this context refers to identifying precise measurable quantities and mathematically describing their patterns, the way it is done in physics and engineering, rather than necessarily proving rigorous theorems. Given the societal impact of rapidly proliferating AI, developing such a theory and applying its analyses to real systems is a compelling and urgent need.

Equilibration of isolated systems: Investigating the role of coarse-graining on the initial state magnetization

Many theoretical and experimental results show that even isolated quantum systems evolving unitarily may equilibrate, since the evolution of some observables may be around an equilibrium value with negligible fluctuations most of the time. There are rigorous theorems giving the conditions for such equilibration to happen. In particular, initial states prepared with a lack of resolution in the energy will equilibrate. We investigate how equilibration may be affected by a lack of resolution, or coarse-graining, in the magnetization of the initial state. In particular, for a chaotic spin chain and using exact diagonalization, we show that the level of equilibration of an initial state with a coarse, not well-defined magnetization is different from the level of an initial state with well-defined magnetization. This difference will depend on the degree of coarse-graining and the direction of magnetization. We also analyze the time for the system to reach equilibrium, showing good agreement with theoretical estimates and with some evidence that less resolution leads to faster equilibration. Our study highlights the crucial role of initial state preparation in the equilibration dynamics of quantum systems and provides new insights into the fundamental nature of equilibration in closed systems.

From statistical physics to social sciences: the pitfalls of multi-disciplinarity

This is the English version of my inaugural lecture at Collège de France in 2021. I reflect on the difficulty of multi-disciplinary research, which often hinges on unexpected epistemological and methodological differences, for example about the scientific status of models. What is the purpose of a model? What are we ultimately trying to establish: rigorous theorems or ad-hoc calculation recipes; absolute truth, or heuristic representations of the world? I argue that the main contribution of statistical physics to social and economic sciences is to make us realise that unexpected behaviour can emerge at the aggregate level, that isolated individuals would never experience. Crises, panics, opinion reversals, the spread of rumours or beliefs, fashion effects and the zeitgeist, but also the existence of money, lasting institutions, social norms and stable societies, must be understood in terms of collective belief and/or trust, self-sustained by interactions, or on the contrary, the rapid collapse of this belief or trust. The appendix contains my opening remarks to the workshop ‘More is Different’, as a tribute to Phil Anderson.

Causal potency of consciousness in the physical world

The evolution of the human mind through natural selection mandates that our conscious experiences are causally potent in order to leave a tangible impact on the surrounding physical world. Any attempt to construct a functional theory of the conscious mind within the framework of classical physics, however, inevitably leads to causally impotent conscious experiences in direct contradiction to evolution theory. Here, we derive several rigorous theorems that identify the origin of the latter impasse in the mathematical properties of ordinary differential equations employed in combination with the alleged functional production of the mind by the brain. Then, we demonstrate that a mind–brain theory consistent with causally potent conscious experiences is provided by modern quantum physics, in which the unobservable conscious mind is reductively identified with the quantum state of the brain and the observable brain is constructed by the physical measurement of quantum brain observables. The resulting quantum stochastic dynamics obtained from sequential quantum measurements of the brain is governed by stochastic differential equations, which permit genuine free will exercised through sequential conscious choices of future courses of action. Thus, quantum reductionism provides a solid theoretical foundation for the causal potency of consciousness, free will and cultural transmission.

Optimal Robust Tracking Control of Injection Velocity in an Injection Molding Machine

Injection molding is a critical component of modern industrial operations, and achieving fast and stable control of injection molding machines (IMMs) is essential for producing high-quality plastic products. This paper focuses on solving an optimal tracking control problem of the injection velocity that arises in a typical nonlinear IMM. To this end, an efficient optimal robust controller is proposed and designed. The nonlinear injection velocity servo system is first approximately linearized at iteration points using the first-order Taylor expansion approach. Then, at each time node in the optimization process, the relevant algebraic Riccati equation is introduced, and the solution is used to construct an optimal robust feedback controller. Furthermore, a rigorous Lyapunov theorem analysis is employed to demonstrate the global stability properties of the proposed feedback controller. The results from numerical simulations show that the proposed optimal robust control strategy can successfully and rapidly achieve the best tracking of the intended injection velocity trajectory within a given time.

Equilibrium investment with random risk aversion.

We solve the problem of an investor who maximizes utility but faces random preferences. We propose a problem formulation based on expected certainty equivalents. We tackle the time-consistency issues arising from that formulation by applying the equilibrium theory approach. To this end, we provide the proper definitions and prove a rigorous verification theorem. We complete the calculations for the cases of power and exponential utility. For power utility, we illustrate in a numerical example that the equilibrium stock proportion is independent of wealth, but decreasing in time, which we also supplement by a theoretical discussion. For exponential utility, the usual constant absolute risk aversion is replaced by itsexpectation.

Downward quantum learning from element 118: Automated generation of Fermi-Löwdin orbitals for all atoms.

A new algorithm based on a rigorous theorem and quantum data computationally mined from element 118 guarantees automated construction of initial Fermi-Löwdin-Orbital (FLO) starting points for all elements in the Periodic Table. It defines a means for constructing a small library of scalable FLOs for universal use in molecular and solid-state calculations. The method can be systematically improved for greater efficiency and for applications to excited states such as x-ray excitations and optically silent excitations. FLOs were introduced to recast the Perdew-Zunger self-interaction correction (PZSIC) into an explicit unitarily invariant form. The FLOs are generated from a set of N quasi-classical electron positions, referred to as Fermi-Orbital descriptors (FODs), and a set of N-orthonormal single-electron orbitals. FOD positions, when optimized, minimize the PZSIC total energy. However, creating sets of starting FODs that lead to a positive definite Fermi orbital overlap matrix has proven to be challenging for systems composed of open-shell atoms and ions. The proof herein guarantees the existence of a FLOSIC solution and further guarantees that if a solution for N electrons is found, it can be used to generate a minimum of N - 1 and a maximum of 2N - 2 initial starting points for systems composed of a smaller number of electrons. Applications to heavy and super-heavy atoms are presented. All starting solutions reported here were obtained from a solution for element 118, Oganesson.

Partial Identifiability for Nonnegative Matrix Factorization

Given a nonnegative matrix factorization, , and a factorization rank, , exact nonnegative matrix factorization (exact NMF) decomposes as the product of two nonnegative matrices, and with columns, such as . A central research topic in the literature is the conditions under which such a decomposition is unique/identifiable up to trivial ambiguities. In this paper, we focus on partial identifiability, that is, the uniqueness of a subset of columns of and . We start our investigations with the data‐based uniqueness (DBU) theorem from the chemometrics literature. The DBU theorem analyzes all feasible solutions of exact NMF and relies on sparsity conditions on and . We provide a mathematically rigorous theorem of a recently published restricted version of the DBU theorem, relying only on simple sparsity and algebraic conditions: it applies to a particular solution of exact NMF (as opposed to all feasible solutions) and allows us to guarantee the partial uniqueness of a single column of or . Second, based on a geometric interpretation of the restricted DBU theorem, we obtain a new partial identifiability result. This geometric interpretation also leads us to another partial identifiability result in the case . Third, we show how partial identifiability results can be used sequentially to guarantee the identifiability of more columns of and . We illustrate these results on several examples, including one from the chemometrics literature.

Application of Carrillo's method in unsaturated soil consolidation

The feasibility of Carrillo's method is confirmed for consolidation problems of unsaturated soils under instantaneous loading in this study. In the plane strain and axial symmetry consolidation models, the coupled governing equations expressed by the excess pore-air and pore-water pressures are transformed into the equivalent uncoupled equations expressed by two functions of excess pore pressure using a decoupled method. Then, two rigorous theorems are used to prove that Carrillo's method is suitable for unsaturated soil consolidation under the plane strain and axial symmetry conditions. Finally, the mathematical formulas by Carrillo's method for unsaturated soil consolidation under plane strain and axisymmetric conditions are presented, and are verified against those of the existing analytical solutions. It turns out that the combinations of existing solutions to the conditions of a single direction flow by the proposed formulas can directly furnish the solutions to unsaturated soil consolidation with both horizontal and vertical flows.

Formation tracking control design for uncertain artificial swarm systems under inequality constraints: Leakage and dead-zone type

The paper proposes a formation tracking control method for the uncertain artificial swarm systems under the inequality constraints. Not only can the agents perform swarm behaviors (e.g., convergence, formation and avoidance of collision), but they can also track the fixed targets in a constrained area (which is formulated as the inequality constraints, such as unilateral constraint and bilateral constraint.). The swarm behaviors are creatively considered as the servo constraints or the control objectives for the swarm agents. Based on the Udwadia–Kalaba (U–K) equation, those prescribed behaviors are realized by a model-based control design (that is the servo constraints force model-based feedforward control). To deal with the inequality constraints in the formation tracking process, a differential homeomorphism transformation is used to relieve the environmental constraints for the swarm agents. Moreover, the uncertainty of the swarm agents (i.e., the parameter uncertainty in modeling and the external disturbances) is considered, which is time-varying and unknown (but bounded). An uncertainty estimation method with dead-zone and leakage term is designed to calculate the possible upper bound of the uncertainty. In virtue of the estimated upper bound of the uncertainty, a robust control is designed for the uncertain swarm agents to obey the prescribed swarm behaviors in the formation tracking task. The system performances of the artificial swarm systems under the proposed control are theoretically guaranteed by a range of rigorous theorems and numerically verified by the simulations of three agents.

Revisiting the averaged problem in the case of mean-motion resonances in the restricted three-body problem

A classical approach to the restricted three-body problem is to analyze the dynamics of the massless body in the synodic reference frame. A different approach is represented by the perturbative treatment: in particular the averaged problem of a mean-motion resonance allows to investigate the long-term behavior of the solutions through a suitable approximation that focuses on a particular region of the phase space. In this paper, we intend to bridge a gap between the two approaches in the specific case of mean-motion resonant dynamics, establish the limit of validity of the averaged problem and take advantage of its results in order to compute trajectories in the synodic reference frame. After the description of each approach, we develop a rigorous treatment of the averaging process, estimate the size of the transformation and prove that the averaged problem is a suitable approximation of the restricted three-body problem as long as the solutions are located outside the Hill’s sphere of the secondary. In such a case, a rigorous theorem of stability over finite but large timescales can be proven. We establish that a solution of the averaged problem provides an accurate approximation of the trajectories on the synodic reference frame within a finite time that depend on the minimal distance to the Hill’s sphere of the secondary. The last part of this work is devoted to the co-orbital motion (i.e., the dynamics in 1:1 mean-motion resonance) in the circular-planar case. In this case, an interpretation of the solutions of the averaged problem in the synodic reference frame is detailed and a method that allows to compute co-orbital trajectories is displayed.

Signal Analysis via the Stochastic Geometry of Spectrogram Level Sets

Spectrograms are fundamental tools in time-frequency analysis, being the squared magnitude of the so-called short time Fourier transform (STFT). Signal analysis via spectrograms has traditionally explored their peaks, i.e. their maxima. This is complemented by a recent interest in their zeros or minima, following seminal work by Flandrin and others, which exploits connections with Gaussian analytic functions (GAFs). However, the zero sets (or extrema) of GAFs have a complicated stochastic structure, complicating any direct theoretical analysis. Standard techniques largely rely on statistical observables from the analysis of spatial data, whose distributional properties for spectrograms are mostly understood only at an empirical level. In this work, we investigate spectrogram analysis via an examination of the stochastic geometric properties of their level sets. We obtain rigorous theorems demonstrating the efficacy of a spectrogram level sets based approach to the detection and estimation of signals, framed in a concrete inferential set-up. Exploiting these ideas as theoretical underpinnings, we propose a level sets based algorithm for signal analysis that is intrinsic to given spectrogram data, and substantiate its effectiveness via extensive empirical studies. Our results also have theoretical implications for spectrogram zero based approaches to signal analysis. To our knowledge, these results are arguably among the first to provide a rigorous statistical understanding of signal detection and reconstruction in this set up, complemented with provable guarantees on detection thresholds and rates of convergence.

Does the second critical-point of water really exist in nature?

In the past decade, a literary phrase “No man's land” has been flooded in the scientific papers. The expression is used to describe a meta-stable region in the phase-diagram that cannot be accessed by experiments. It has been claimed based on the molecular dynamics (MD) simulation that there is a critical point, or the second critical point (SCP), in the “no man's land,” and it has created a big dispute in the field of science. It is proved in the present paper that the hypothesis of SCP is completely against the rigorous theorem of thermodynamics, referred as the Gibbs phase rule. The reason why the simulations have found SCP erroneously is merely because the method violates the requirement which all the statistical-mechanics treatments should satisfy to reproduce the thermodynamics. That is the thermodynamic limit. It is clarified what is the identity of the ``liquid-liquid phase transition'' and SCP in pure liquids, discovered by the simulations and by some experiments. In order to explain the physics of liquid-liquid phase transition observed experimentally in single component liquids, a new concept is proposed.

Data‐driven optimal fault‐tolerant‐control and detection for a class of unknown nonlinear discrete‐time systems

AbstractThe design of optimal fault‐tolerant control with actuator and sensor faults is investigated. The system dynamics and faults are considered as a class of unknown nonlinear discrete‐time systems when the data‐driven equivalent model is formulated by a multi‐input fuzzy rule emulated network (MiFREN). The multi‐gradient learning law is developed with the proposed fault‐detection algorithm to enchant the performance of MiFREN. By employing only pieces of information from the equivalent model, the proposed controller is designed and the closed‐loop performance is analyzed through the rigorous theorem. Simulation systems and comparison results are provided to validate the performance of the proposed scheme.

Chaotic vibrations of 3D linear hyperbolic PDEs with linear perturbations of superlinear boundary conditions

This article studies the chaotic vibrations of the infinite-dimensional dynamical systems governed by linear hyperbolic partial differential equations (PDEs) in three-dimensional (3D) space, where the boundary conditions include two linear perturbations of superlinear types. A rigorous mathematical theorem that guarantees the occurrence of chaos of such systems is obtained. The main theme of this article is the advancement of existing chaos to 3D hyperbolic PDEs. As applications, two examples are provided for showing the effectiveness of the theoretical chaotic results.

Frictional contact problem of a rigid charged indenter on two dimensional hexagonal piezoelectric quasicrystals coating

ABSTRACT This paper reports the frictional contact problem of a rigid charged indenter on two-dimensional (2D) hexagonal piezoelectric quasicrystals (PEQCs) coating. Using the rigorous operator theory and Almansi theorem, two sets of three-dimensional (3D) general solutions for 2D-hexagonal PEQCs are obtained. The frequency response functions (FRFs) for displacements and stresses of phonon and phason, electric potentials and electric displacements, which are the multifield responses of material under generalised unit point load in the Fourier domain, are obtained by applying double Fourier integral transforms to the general solutions and boundary conditions. The contact responses, such as pressures and stresses, are solved by the discrete convolution-fast Fourier transform technique (DC-FFT). Numerical results are given to reveal the influences of material parameters and loading conditions on the contact behaviour. The obtained 3D contact solutions are not only helpful further analysis and understanding of the coupling characteristics of phonon, phason and electric field, but also provide a reference basis for experimental analysis and material development.

Optimal Investment with Uncertain Risk Aversion

We solve the problem of an investor who maximizes utility but is uncertain about preferences. We propose a problem formulation based on expected certainty equivalents. We tackle the time-consistency issues arising from that formulation by applying the equilibrium theory approach. To this end, we provide the proper de nitions and proof a rigorous veri fication theorem. We complete the calculations for the cases of power and exponential utility. For power utility, we illustrate in a numerical example, that the optimal stock proportion is independent of wealth, but decreasing in time. For exponential utility, the usual constant absolute risk aversion is replaced by its expectation.

Structural Analysis of Large-Scale Socio-Technical Systems Based on the Concept of Influence

Abstract The paper is devoted to studying the stability of complex systems, which include diversified and heterogeneous elements. In order to analyze the stability of such systems, it is proposed to develop the process of dynamic determination of the main factors. For this purpose, this paper uses a variant of cognitive modeling based on the concept of “influence”. The proposed approach implies, in a sense, a broader view of the concept of the “fuzzy cognitive map” introduced by B. Kosko. “Influence” does not mean “causality”, allows a broader interpretation range, and is calculated in a special way that makes it possible to prove rigorous theorems. A complex system is represented as a digraph, and the selection of the main factors is proposed to be based on the concept of “influence”, which is introduced as follows. All nodes of the graph are assigned an abstract property “significance”, which allows you to compare heterogeneous factors, and a particular computational procedure is introduced that allows this “significance” to be calculated. The “influence” of factors on each other is defined as a mathematically formalized response in accordance with the introduced computational procedure of the “significance” of the studied factor to the variation in the “significance” of input factors. For example, the analysis of the sustainability of a large-scale model of the Russian economy in terms of the strength of the “influence” is provided.

Noise resilience of variational quantum compiling

Variational hybrid quantum-classical algorithms (VHQCAs) are near-term algorithms that leverage classical optimization to minimize a cost function, which is efficiently evaluated on a quantum computer. Recently VHQCAs have been proposed for quantum compiling, where a target unitary U is compiled into a short-depth gate sequence V. In this work, we report on a surprising form of noise resilience for these algorithms. Namely, we find one often learns the correct gate sequence V (i.e. the correct variational parameters) despite various sources of incoherent noise acting during the cost-evaluation circuit. Our main results are rigorous theorems stating that the optimal variational parameters are unaffected by a broad class of noise models, such as measurement noise, gate noise, and Pauli channel noise. Furthermore, our numerical implementations on IBM’s noisy simulator demonstrate resilience when compiling the quantum Fourier transform, Toffoli gate, and W-state preparation. Hence, variational quantum compiling, due to its robustness, could be practically useful for noisy intermediate-scale quantum devices. Finally, we speculate that this noise resilience may be a general phenomenon that applies to other VHQCAs such as the variational quantum eigensolver.

Accurately Predicting the Radiation Enhancement Factor in Plasmonic Optical Antenna Emitters.

Plasmonic optical antennas (POAs), often constructed from gold or silver nanostructures, can enhance the radiation efficiency of emitters coupled to POAs and are applied in surface-enhanced Raman spectroscopy (SERS) and light-emitting devices. Over the past four decades, radiation enhancement factors (REFs) of POA-emitter systems were considered to be difficult to calculate directly and have been predicted indirectly and approximately, assuming POAs are illuminated by electromagnetic plane waves without emitters. The validity of this approximation remains a significant open problem in SERS theory. Herein, we develop a method based on the rigorous optical reciprocity theorem for accurately calculating the REFs of emitters in nanoparticle-substrate nanogaps for single-molecule SERS and scanning probe-substrate nanogaps for tip-enhanced Raman spectroscopy. We show that the validity of the plane wave approximation breaks down if high-order plasmonic modes are excited. The as-developed method paves the way toward designing high-REF POA nanostructures for luminescence-related devices.