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We introduce a proof-theoretic method for showing nondefinability of second-order intuitionistic connectives by quantifier-free schemata. We apply the method to confirm that Taranovsky's "realizability disjunction" connective does not admit a quantifier-free definition, and use it to obtain new results and more nuanced information about the nondefinability of Kreisel's and Po{\l}acik's unary connectives. The finitary and combinatorial nature of our method makes it more resilient to changes in metatheory than common semantic approaches, whose robustness tends to waver once we pass to non-classical and especially anti-classical settings. Furthermore, we can easily transcribe the problem-specific subproofs into univalent type theory and check them using the Agda proof assistant.

Deep learning-based numerical schemes for solving high-dimensional backward stochastic differential equations (BSDEs) have recently raised plenty of scientific interest. While they enable numerical methods to approximate very high-dimensional BSDEs, their reliability has not been studied and is thus not understood. In this work, we study uncertainty quantification (UQ) for a class of deep learning-based BSDE schemes. More precisely, we review the sources of uncertainty involved in the schemes and numerically study the impact of different sources. Usually, the standard deviation (STD) of the approximate solutions obtained from multiple runs of the algorithm with different datasets is calculated to address the uncertainty. This approach is computationally quite expensive, especially for high-dimensional problems. Hence, we develop a UQ model that efficiently estimates the STD of the approximate solution using only a single run of the algorithm. The model also estimates the mean of the approximate solution, which can be leveraged to initialize the algorithm and improve the optimization process. Our numerical experiments show that the UQ model produces reliable estimates of the mean and STD of the approximate solution for the considered class of deep learning-based BSDE schemes. The estimated STD captures multiple sources of uncertainty, demonstrating its effectiveness in quantifying the uncertainty. Additionally, the model illustrates the improved performance when comparing different schemes based on the estimated STD values. Furthermore, it can identify hyperparameter values for which the scheme achieves good approximations.

We present a priori error estimates for a multirate time-stepping scheme for coupled differential equations. The discretization is based on Galerkin methods in time using two different time meshes for two parts of the problem. We aim at surface coupled multiphysics problems like two-phase flows. Special focus is on the handling of the interface coupling to guarantee a coercive formulation as key to optimal order error estimates. In a sequence of increasing complexity, we begin with the coupling of two ordinary differential equations, coupled heat conduction equation, and finally a coupled Stokes problem. For this we show optimal multi-rate estimates in velocity and a suboptimal result in pressure. The a priori estimates prove that the multirate method decouples the two subproblems exactly. This is the basis for adaptive methods which can choose optimal lattices for the respective subproblems.

High-quality samples generated with score-based reverse diffusion algorithms provide evidence that deep neural networks (DNN) trained for denoising can learn high-dimensional densities, despite the curse of dimensionality. However, recent reports of memorization of the training set raise the question of whether these networks are learning the "true" continuous density of the data. Here, we show that two denoising DNNs trained on non-overlapping subsets of a dataset learn nearly the same score function, and thus the same density, with a surprisingly small number of training images. This strong generalization demonstrates an alignment of powerful inductive biases in the DNN architecture and/or training algorithm with properties of the data distribution. We analyze these, demonstrating that the denoiser performs a shrinkage operation in a basis adapted to the underlying image. Examination of these bases reveals oscillating harmonic structures along contours and in homogeneous image regions. We show that trained denoisers are inductively biased towards these geometry-adaptive harmonic representations by demonstrating that they arise even when the network is trained on image classes such as low-dimensional manifolds, for which the harmonic basis is suboptimal. Additionally, we show that the denoising performance of the networks is near-optimal when trained on regular image classes for which the optimal basis is known to be geometry-adaptive and harmonic.

The equioscillation condition is extended to multivariate approximation. To this end, it is reformulated as the synchronized oscillations between the error maximizers and the components of a related Haar matrix kernel vector. This new condition gives rise to a multivariate equioscillation theorem where the Haar condition is not assumed and hence the existence and the characterization by equioscillation become independent of uniqueness. This allows the theorem to be applicable to problems with no strong uniqueness or even no uniqueness. A technical additional requirement on the involved Haar matrix and its kernel vector is proved to be sufficient for strong uniqueness. Instances of multivariate problems with strongly unique, unique and nonunique solutions are presented to illustrate the scope of the theorem.

We study in this paper the monotonicity properties of the numerical solutions to Volterra integral equations with nonincreasing completely positive kernels on nonuniform meshes. There is a duality between the complete positivity and the properties of the complementary kernel being nonnegative and nonincreasing. Based on this, we propose the ``complementary monotonicity'' to describe the nonincreasing completely positive kernels, and the ``right complementary monotone'' (R-CMM) kernels as the analogue for nonuniform meshes. We then establish the monotonicity properties of the numerical solutions inherited from the continuous equation if the discretization has the R-CMM property. Such a property seems weaker than being log-convex and there is no resctriction on the step size ratio of the discretization for the R-CMM property to hold.

Hamilton-Jacobi (HJ) partial differential equations (PDEs) have diverse applications spanning physics, optimal control, game theory, and imaging sciences. This research introduces a first-order optimization-based technique for HJ PDEs, which formulates the time-implicit update of HJ PDEs as saddle point problems. We remark that the saddle point formulation for HJ equations is aligned with the primal-dual formulation of optimal transport and potential mean-field games (MFGs). This connection enables us to extend MFG techniques and design numerical schemes for solving HJ PDEs. We employ the primal-dual hybrid gradient (PDHG) method to solve the saddle point problems, benefiting from the simple structures that enable fast computations in updates. Remarkably, the method caters to a broader range of Hamiltonians, encompassing non-smooth and spatiotemporally dependent cases. The approach's effectiveness is verified through various numerical examples in both one-dimensional and two-dimensional examples, such as quadratic and $L^1$ Hamiltonians with spatial and time dependence.

We construct and analyze a message-passing algorithm for random constraint satisfaction problems (CSPs) at large clause density, generalizing work of El Alaoui, Montanari, and Sellke for Maximum Cut [arXiv:2111.06813] through a connection between random CSPs and mean-field Ising spin glasses. For CSPs with even predicates, the algorithm asymptotically solves a stochastic optimal control problem dual to an extended Parisi variational principle. This gives an optimal fraction of satisfied constraints among algorithms obstructed by the branching overlap gap property of Huang and Sellke [arXiv:2110.07847], notably including the Quantum Approximate Optimization Algorithm and all quantum circuits on a bounded-degree architecture of up to $\epsilon \cdot \log n$ depth.

This article is concerned with a regularity analysis of parametric operator equations with a perspective on uncertainty quantification. We study the regularity of mappings between Banach spaces near branches of isolated solutions that are implicitly defined by a residual equation. Under $s$-Gevrey assumptions on on the residual equation, we establish $s$-Gevrey bounds on the Fr\'echet derivatives of the local data-to-solution mapping. This abstract framework is illustrated in a proof of regularity bounds for a semilinear elliptic partial differential equation with parametric and random field input.

We derive information-theoretic generalization bounds for supervised learning algorithms based on the information contained in predictions rather than in the output of the training algorithm. These bounds improve over the existing information-theoretic bounds, are applicable to a wider range of algorithms, and solve two key challenges: (a) they give meaningful results for deterministic algorithms and (b) they are significantly easier to estimate. We show experimentally that the proposed bounds closely follow the generalization gap in practical scenarios for deep learning.