Learning Interaction Kernels in Stochastic Systems of Interacting Particles from Multiple Trajectories

We consider stochastic systems of interacting particles or agents, with dynamics determined by an interaction kernel, which only depends on pairwise distances. We study the problem of inferring this interaction kernel from observations of the positions of the particles, in either continuous or discrete time, along multiple independent trajectories. We introduce a nonparametric inference approach to this inverse problem, based on a regularized maximum likelihood estimator constrained to suitable hypothesis spaces adaptive to data. We show that a coercivity condition enables us to control the condition number of this problem and prove the consistency of our estimator, and that in fact it converges at a near-optimal learning rate, equal to the min–max rate of one-dimensional nonparametric regression. In particular, this rate is independent of the dimension of the state space, which is typically very high. We also analyze the discretization errors in the case of discrete-time observations, showing that it is of order 1/2 in terms of the time spacings between observations. This term, when large, dominates the sampling error and the approximation error, preventing convergence of the estimator. Finally, we exhibit an efficient parallel algorithm to construct the estimator from data, and we demonstrate the effectiveness of our algorithm with numerical tests on prototype systems including stochastic opinion dynamics and a Lennard-Jones model.

A Deep Learning Approach for Recognizing Activity of Daily Living (ADL) for Senior Care: Exploiting Interaction Dependency and Temporal Patterns

Ensuring the health and safety of senior citizens who live alone is a growing societal concern. The Activity of Daily Living (ADL) approach is a common means to monitor disease progression and the ability of these individuals to care for themselves. However, the prevailing sensor-based ADL monitoring systems primarily rely on wearable motion sensors, capture insufficient information for accurate ADL recognition, and do not provide a comprehensive understanding of ADLs at different granularities. Current healthcare IS and mobile analytics research focuses on studying the system, device, and provided services, and is in need of an end-to-end solution to comprehensively recognize ADLs based on mobile sensor data. This study adopts the design science paradigm and employs advanced deep learning algorithms to develop a novel hierarchical, multiphase ADL recognition framework to model ADLs at different granularities. We propose a novel 2D interaction kernel for convolutional neural networks to leverage interactions between human and object motion sensors. We rigorously evaluate each proposed module and the entire framework against state-of-the-art benchmarks (e.g., support vector machines, DeepConvLSTM, hidden Markov models, and topic-modeling-based ADLR) on two real-life motion sensor datasets that consist of ADLs at varying granularities: Opportunity and INTER. Results and a case study demonstrate that our framework can recognize ADLs at different levels more accurately. We discuss how stakeholders can further benefit from our proposed framework. Beyond demonstrating practical utility, we discuss contributions to the IS knowledge base for future design science-based cybersecurity, healthcare, and mobile analytics applications.

Deriving the dark matter-dark energy interaction term in the continuity equation from the Boltzmann equation

Dark energy and dark matter are two of the biggest mysteries of modern cosmology, and our understanding of their fundamental nature is incomplete. Many parametrizations of couplings between the two in the continuity equation have been studied in the literature, and observational data from the growth of perturbations can constrain these parametrizations. Assuming standard general relativity with a simple Yukawa-type coupling between dark energy and dark matter fields in the Lagrangian, we use the Boltzmann equation to analytically express and calculate the interaction kernel Q in the continuity equation and compare it to that of a typical parametrization. We arrive at a comparably very small result, as expected. Since the interaction is a function of the dark matter mass, other observational data sets can be used to constrain the mass. This calculation can be modified to account for other couplings of the dark energy and dark matter fields. This calculation required obtaining a distribution function for dark energy that leads to an equation of state parameter that is negative, which neither Bose–Einstein nor Fermi–Dirac statistics can supply, and this is the main result of this paper. Treating dark energy as a quantum scalar field, we use adiabatic subtraction to obtain a finite analytic approximation for its distribution function that assumes the FLRW metric and nothing more.

The equilibrium measure for an anisotropic nonlocal energy

In this paper we characterise the minimisers of a one-parameter family of nonlocal and anisotropic energies $$I_\alpha $$ I α defined on probability measures in $${\mathbb {R}}^n$$ R n , with $$n\ge 3$$ n ≥ 3 . The energy $$I_\alpha $$ I α consists of a purely nonlocal term of convolution type, whose interaction kernel reduces to the Coulomb potential for $$\alpha =0$$ α = 0 and is anisotropic otherwise, and a quadratic confinement. The two-dimensional case arises in the study of defects in metals and has been solved by the authors by means of complex-analysis techniques. We prove that for $$\alpha \in (-1, n-2]$$ α ∈ ( - 1 , n - 2 ] , the minimiser of $$I_\alpha $$ I α is unique and is the (normalised) characteristic function of a spheroid. This result is a paradigmatic example of the role of the anisotropy of the kernel on the shape of minimisers. In particular, the phenomenon of loss of dimensionality, observed in dimension $$n=2$$ n = 2 , does not occur in higher dimension at the value $$\alpha =n-2$$ α = n - 2 corresponding to the sign change of the Fourier transform of the interaction potential.

Tetraquark state X(6900) and the interaction between diquark and antidiquark

Recently LHCb declared a new structure X(6900) in the final state di-$$J/\psi $$ J / ψ which is popularly regarded as a cc-$$\bar{c}\bar{c}$$ c ¯ c ¯ tetraquark state. Within the Bethe–Salpeter (B–S) framework we study the possible cc-$$\bar{c}\bar{c}$$ c ¯ c ¯ bound states and the interaction between diquark (cc) and antidiquark ($$\bar{c}\bar{c}$$ c ¯ c ¯ ). In this work cc ($$\bar{c}\bar{c}$$ c ¯ c ¯ ) is treated as a color anti-triplet (triplet) axial-vector so the quantum numbers of cc-$$\bar{c}\bar{c}$$ c ¯ c ¯ bound state are $$0^+$$ 0 + , $$1^+$$ 1 + and $$2^+$$ 2 + . Learning from the interaction in meson case and using the effective coupling we suggest the interaction kernel for the diquark and antidiquark system. Then we deduce the B–S equations for different quantum numbers. Solving these equations numerically we find the spectra of some excited states can be close to the mass of X(6900) when we assign appropriate values for parameter $$\kappa $$ κ introduced in the interaction (kernel). We also briefly calculate the spectra of bb-$$\bar{b}\bar{b}$$ b ¯ b ¯ bound states. Future measurement of bb-$$\bar{b}\bar{b}$$ b ¯ b ¯ state will help us to determine the exact form of effective interaction.

Landscape-induced spatial oscillations in population dynamics

We study the effect that disturbances in the ecological landscape exert on the spatial distribution of a population that evolves according to the nonlocal FKPP equation. Using both numerical and analytical techniques, we characterize, as a function of the interaction kernel, the three types of stationary profiles that can develop near abrupt spatial variations in the environmental conditions vital for population growth: sustained oscillations, decaying oscillations and exponential relaxation towards a flat profile. Through the mapping between the features of the induced wrinkles and the shape of the interaction kernel, we discuss how heterogeneities can reveal information that would be hidden in a flat landscape.

Macroscopic descriptions of follower-leader systems

<p style='text-indent:20px;'>The fundamental derivation of macroscopic model equations to describe swarms based on microscopic movement laws and mathematical analyses into their self-organisation capabilities remains a challenge from the perspective of both modelling and analysis. In this paper we clarify relevant continuous macroscopic model equations that describe follower-leader interactions for a swarm where these two populations are fixed. We study the behaviour of the swarm over long and short time scales to shed light on the number of leaders needed to initiate swarm movement, according to the homogeneous or inhomogeneous nature of the interaction (alignment) kernel. The results indicate the crucial role played by the interaction kernel to model transient behaviour.</p>

Asymptotic limit of a spatially-extended mean-field FitzHugh–Nagumo model

We consider a spatially extended mean-field model of a FitzHugh–Nagumo neural network, with a rescaled interaction kernel. Our main purpose is to prove that its asymptotic limit in the regime of strong local interactions converges toward a system of reaction–diffusion equations taking account for the average quantities of the network. Our approach is based on a modulated energy argument, to compare the macroscopic quantities computed from the solution of the transport equation, and the solution of the limit system. The main difficulty, compared to the literature, lies in the need of regularity in space of the solutions of the limit system and a careful control of an internal nonlocal dissipation.