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It has long been believed that the brain is highly modular both in terms of structure and function, although recent evidence has led some to question the extent of both types of modularity. We used artificial neural networks to test the hypothesis that structural modularity is sufficient to guarantee functional specialization, and find that in general, this doesn't necessarily hold except at extreme levels. We then systematically tested which features of the environment and network do lead to the emergence of specialization. We used a simple toy environment, task and network, allowing us precise control, and show that in this setup, several distinct measures of specialization give qualitatively similar results. We further find that (1) specialization can only emerge in environments where features of that environment are meaningfully separable, (2) specialization preferentially emerges when the network is strongly resource-constrained, and (3) these findings are qualitatively similar across different network architectures, but the quantitative relationships depends on the architecture type. Finally, we show that functional specialization varies dynamically across time, and demonstrate that these dynamics depend on both the timing and bandwidth of information flow in the network. We conclude that a static notion of specialization, based on structural modularity, is likely too simple a framework for understanding intelligence in situations of real-world complexity, from biology to brain-inspired neuromorphic systems. We propose that thoroughly stress testing candidate definitions of functional modularity in simplified scenarios before extending to more complex data, network models and electrophysiological recordings is likely to be a fruitful approach.

Distinguishing two classes of candidate models is a fundamental and practically important problem in statistical inference. Error rate control is crucial to the logic but, in complex nonparametric settings, such guarantees can be difficult to achieve, especially when the stopping rule that determines the data collection process is not available. In this paper we develop a novel e-process construction that leverages the so-called predictive recursion (PR) algorithm designed to rapidly and recursively fit nonparametric mixture models. The resulting PRe-process affords anytime valid inference uniformly over stopping rules and is shown to be efficient in the sense that it achieves the maximal growth rate under the alternative relative to the mixture model being fit by PR. In the special case of testing for a log-concave density, the PRe-process test is computationally simpler and faster, more stable, and no less efficient compared to a recently proposed anytime valid test.

Network motifs are recurrent, small-scale patterns of interactions observed frequently in a system. They shed light on the interplay between the topology and the dynamics of complex networks across various domains. In this work, we focus on the problem of counting occurrences of small sub-hypergraph patterns in very large hypergraphs, where higher-order interactions connect arbitrary numbers of system units. We show how directly exploiting higher-order structures speeds up the counting process compared to traditional data mining techniques for exact motif discovery. Moreover, with hyperedge sampling, performance is further improved at the cost of small errors in the estimation of motif frequency. We evaluate our method on several real-world datasets describing face-to-face interactions, co-authorship and human communication. We show that our approximated algorithm allows us to extract higher-order motifs faster and on a larger scale, beyond the computational limits of an exact approach.

We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learn the transfer operator of the system, that in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach is supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the context of transfer operator regression. The objective function we propose is associated with projection operators from the representation space to the data space, overcomes metric distortion, and can be empirically estimated from data. In the discrete time setting, we further derive a relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.

We investigate a class of parametric elliptic semilinear partial differential equations of second order with homogeneous essential boundary conditions, where the coefficients and the right-hand side (and hence the solution) may depend on a parameter. This model can be seen as a reaction-diffusion problem with a polynomial nonlinearity in the reaction term. The efficiency of various numerical approximations across the entire parameter space is closely related to the regularity of the solution with respect to the parameter. We show that if the coefficients and the right-hand side are analytic or Gevrey class regular with respect to the parameter, the same type of parametric regularity is valid for the solution. The key ingredient of the proof is the combination of the alternative-to-factorial technique from our previous work [1] with a novel argument for the treatment of the power-type nonlinearity in the reaction term. As an application of this abstract result, we obtain rigorous convergence estimates for numerical integration of semilinear reaction-diffusion problems with random coefficients using Gaussian and Quasi-Monte Carlo quadrature. Our theoretical findings are confirmed in numerical experiments.

We introduce a flexible method to simultaneously infer both the drift and volatility functions of a discretely observed scalar diffusion. We introduce spline bases to represent these functions and develop a Markov chain Monte Carlo algorithm to infer, a posteriori, the coefficients of these functions in the spline basis. A key innovation is that we use spline bases to model transformed versions of the drift and volatility functions rather than the functions themselves. The output of the algorithm is a posterior sample of plausible drift and volatility functions that are not constrained to any particular parametric family. The flexibility of this approach provides practitioners a powerful investigative tool, allowing them to posit a variety of parametric models to better capture the underlying dynamics of their processes of interest. We illustrate the versatility of our method by applying it to challenging datasets from finance, paleoclimatology, and astrophysics. In view of the parametric diffusion models widely employed in the literature for those examples, some of our results are surprising since they call into question some aspects of these models.

We develop a numerical method for the Westervelt equation, an important equation in nonlinear acoustics, in the form where the attenuation is represented by a class of non-local in time operators. A semi-discretisation in time based on the trapezoidal rule and A-stable convolution quadrature is stated and analysed. Existence and regularity analysis of the continuous equations informs the stability and error analysis of the semi-discrete system. The error analysis includes the consideration of the singularity at $t = 0$ which is addressed by the use of a correction in the numerical scheme. Extensive numerical experiments confirm the theory.

Gaussian processes (GPs) are popular nonparametric statistical models for learning unknown functions and quantifying the spatiotemporal uncertainty in data. Recent works have extended GPs to model scalar and vector quantities distributed over non-Euclidean domains, including smooth manifolds appearing in numerous fields such as computer vision, dynamical systems, and neuroscience. However, these approaches assume that the manifold underlying the data is known, limiting their practical utility. We introduce RVGP, a generalisation of GPs for learning vector signals over latent Riemannian manifolds. Our method uses positional encoding with eigenfunctions of the connection Laplacian, associated with the tangent bundle, readily derived from common graph-based approximation of data. We demonstrate that RVGP possesses global regularity over the manifold, which allows it to super-resolve and inpaint vector fields while preserving singularities. Furthermore, we use RVGP to reconstruct high-density neural dynamics derived from low-density EEG recordings in healthy individuals and Alzheimer's patients. We show that vector field singularities are important disease markers and that their reconstruction leads to a comparable classification accuracy of disease states to high-density recordings. Thus, our method overcomes a significant practical limitation in experimental and clinical applications.

The nonlinear Poisson-Boltzmann equation (NPBE) is an elliptic partial differential equation used in applications such as protein interactions and biophysical chemistry (among many others). It describes the nonlinear electrostatic potential of charged bodies submerged in an ionic solution. The kinetic presence of the solvent molecules introduces randomness to the shape of a protein, and thus a more accurate model that incorporates these random perturbations of the domain is analyzed to compute the statistics of quantities of interest of the solution. When the parameterization of the random perturbations is high-dimensional, this calculation is intractable as it is subject to the curse of dimensionality. However, if the solution of the NPBE varies analytically with respect to the random parameters, the problem becomes amenable to techniques such as sparse grids and deep neural networks. In this paper, we show analyticity of the solution of the NPBE with respect to analytic perturbations of the domain by using the analytic implicit function theorem and the domain mapping method. Previous works have shown analyticity of solutions to linear elliptic equations but not for nonlinear problems. We further show how to derive \emph{a priori} bounds on the size of the region of analyticity. This method is applied to the trypsin molecule to demonstrate that the convergence rates of the quantity of interest are consistent with the analyticity result. Furthermore, the approach developed here is sufficiently general enough to be applied to other nonlinear problems in uncertainty quantification.

We present ReCAT, a recursive composition augmented Transformer that is able to explicitly model hierarchical syntactic structures of raw texts without relying on gold trees during both learning and inference. Existing research along this line restricts data to follow a hierarchical tree structure and thus lacks inter-span communications. To overcome the problem, we propose a novel contextual inside-outside (CIO) layer that learns contextualized representations of spans through bottom-up and top-down passes, where a bottom-up pass forms representations of high-level spans by composing low-level spans, while a top-down pass combines information inside and outside a span. By stacking several CIO layers between the embedding layer and the attention layers in Transformer, the ReCAT model can perform both deep intra-span and deep inter-span interactions, and thus generate multi-grained representations fully contextualized with other spans. Moreover, the CIO layers can be jointly pre-trained with Transformers, making ReCAT enjoy scaling ability, strong performance, and interpretability at the same time. We conduct experiments on various sentence-level and span-level tasks. Evaluation results indicate that ReCAT can significantly outperform vanilla Transformer models on all span-level tasks and baselines that combine recursive networks with Transformers on natural language inference tasks. More interestingly, the hierarchical structures induced by ReCAT exhibit strong consistency with human-annotated syntactic trees, indicating good interpretability brought by the CIO layers.