Developed accurate empirical potentials for Si:H nanowires using multi-fidelity Gaussian process.
problem Accurate modeling of Si:H nanowires using fast but inaccurate empirical potentials and slow but accurate first-principle calculations.
method Employed multi-fidelity Gaussian process regression to integrate low-fidelity empirical potential data with high-fidelity first-principle calculations.
result Demonstrated the accuracy of developed empirical potentials for Si:H nanowires.
A neural network derived from first principles using MaxEnt.
problem Developing a neural network from first principles.
method Derived a neural network using the principle of Maximum Entropy, with linear dimension-reducing transformations and conditional mean estimators.
result Unified theoretical justification for activation functions like sigmoid, softplus, and relu.
Study shows strong min-max principle for phase transitions.
problem Understanding nodal sets near minimal hypersurfaces.
method Analogous to White's principle, applies to Allen-Cahn energy.
result Strong min-max principle for phase transitions.
This paper formalizes the h-principle and sphere eversion in differential topology.
problem Formalizing the h-principle and sphere eversion in differential topology.
method Lean formalization of the local h-principle for first-order partial differential relations, using convex integration.
result Reproves Smale's sphere eversion theorem and formalizes advanced mathematics.
Paper proves h-principles for symplectic structures and foliations.
problem Existence of conformal symplectic structures and foliations.
method Application of h-principles and foliated Morse theory.
result Linear deformation of foliations to contact structures.
We study weak solutions to degenerate quasilinear elliptic equations, involving first order terms, in unbounded tubular domains. In particular we show that, under suitable hypotheses, the weak comparison principle holds if the domain is narrow enough.
Compact Kähler spaces with zero first Chern class have special geometric properties.
problem Characterizing Kähler spaces with zero first Chern class.
method Analyzing holomorphic tensors, decomposing tangent sheaf, and using Bochner principle.
result Spaces with zero first Chern class split off a complex torus and have specific holonomy representations.
Established concavity principle for curved spaces.
problem Solving equations on curved spaces with nonnegative curvature.
method Applied concavity principle to elliptic and parabolic equations on locally symmetric spaces with nonnegative curvature.
result First general concavity principle on spaces with non-constant sectional curvature.
Why deep neural networks (DNNs) capable of overfitting often generalize well in practice is a mystery [#zhang2016understanding]. To find a potential mechanism, we focus on the study of implicit biases underlying the training process of DNNs. In this work, for both real and synthetic datasets, we empirically find that a…
We establish the proportionality principle between the Riemannian volume and locally finite simplicial volume for Q-rank 1 locally symmetric spaces covered by products of hyperbolic spaces, giving the first examples for manifolds whose cusp groups are not necessarily amenable. Also, we give a simple direct proof of the…
Blockchain aims to improve trust in AI systems, but lacks systematic studies.
problem Lack of systematic studies on blockchain design principles for AI trust.
method Hybrid qualitative and quantitative studies.
result Vast opportunities for future research and practice in blockchain design.
Paper explores two methods for optimal portfolio selection in financial markets.
problem Optimal portfolio selection for financial markets with jumps.
method Maximum principle and dynamic programming approach.
result Relationship between two methods and their adjoint processes.
Study the averaging principle for non-autonomous slow-fast systems and apply it to financial local stochastic volatility models.
problem Understanding the behavior of non-autonomous slow-fast systems of stochastic differential equations.
method Prove the averaging principle under specific conditions and apply it to a financial model.
result Prices of derivatives converge to those calculated using the limit model under a risk-neutral measure.
Derives FACT, an alternative to NFA for neural networks, explaining feature learning.
problem Understanding how neural networks learn representations.
method First-principles approach using first-order optimality conditions.
result FACT explains why NFA holds and provides a principled alternative.
Researchers find optimal paths on a specific geometric group.
problem Finding optimal paths on a Cartan group with a sub-Finsler quasimetric.
method Using the Pontryagin Maximum Principle in coordinates of the first kind.
result They found extremals for arbitrary left-invariant sub-Finsler quasimetrics.
We establish large deviation principles for convolutional neural networks.
problem Understanding the behavior of convolutional neural networks in the infinite-channel limit.
method We establish large deviation principles for convolutional neural networks under Gaussian prior and posterior distributions.
result We provide a large deviation principle for the sequence of conditional covariance matrices and the posterior distribution.
New principle for optimal control with higher order differential constraints.
problem Optimal control problems with higher order differential constraints.
method Derivation of the Principle of Minimal Labour and generalization of Pontryagin Maximum Principle.
result Generalized Pontryagin Maximum Principle for higher order constraints.
We discuss the essential self-adjointness of wave operators, as well as the limiting absorption principle, in generalizations of asymptotically Minkowski settings. This is obtained via using a Fredholm framework for inverting the spectral family first, and then refining its conclusions to show its dense range in L^2 wh…
The paper explores variational principles for equations of maximal symmetry, providing new insights and results.
problem Exploring variational principles for equations of maximal symmetry.
method Study of variational and divergence symmetries for linear and nonlinear equations of maximal symmetry, providing first integrals in explicit form.
result Significantly different results and more general variational symmetry algebra for linear and nonlinear equations compared to previous studies.
Derives a new first order differential equation for smooth surfaces.
problem Finding new equations to describe smooth surfaces.
method Derives a linear differential equation of the first order.
result Proves the maximum principle for Darboux rotation fields.
Book introduces principles of LCK geometry for complex manifold students.
problem Understanding locally conformally Kahler manifolds.
method Explains the concept and properties of LCK manifolds.
result Provides insights into the structure and behavior of LCK manifolds.
PRI-VAE learns disentangled representations by optimizing principle-of-relevant-information.
problem Learning disentangled representations under VAE framework remains unknown.
method Proposes PRI-VAE, a novel learning objective to optimize disentanglement.
result Demonstrates effectiveness of PRI-VAE on four benchmark datasets.
Paper proposes hybrid machine learning for tuning first principles models in engineering systems.
problem Inaccurate first principles models in process engineering due to changing conditions.
method Hybrid machine learning framework using Bayesian Neural Networks.
result Uncertainty estimates improve operation decisions in multiphase flow modeling.
The net-premium principle is considered to be the most genuine and fair premium principle in actuarial applications. However, an insurance company, applying the net-premium principle, goes bankrupt with probability one in the long run, even if the company covers its entire costs by collecting the respective fees from i…
Paper confirms Yau's conjecture about sphere eigenvalues.
problem Yau's conjecture on eigenvalues of minimal hypersurfaces.
method Constructing a minimizing sequence in Sobolev space, using variational principle.
result First non-zero eigenvalue equals hypersurface dimension.
In this paper, we use the distance comparison principle, first been developed by G. Huisken, to study the spatial curve shortening flow. We have got the result that if the initial curve is the helix, then the local minimum of the ratio of the extrinsic and intrinsic distance is non-decreasing. And we have proved a Gray…
The minimum description length (MDL) principle in supervised learning is studied. One of the most important theories for the MDL principle is Barron and Cover's theory (BC theory), which gives a mathematical justification of the MDL principle. The original BC theory, however, can be applied to supervised learning only …
New principle optimizes bandit decisions with context.
problem Dealing with general function classes and large context spaces in bandits.
method Upper Counterfactual Confidence Bounds (UCCB) principle.
result Proves optimality and efficiency in complex settings.
Use simplified layerwise linear models to understand neural dynamics.
problem Complex neural network dynamics are hard to grasp.
method Apply simplified layerwise linear models to explain neural phenomena.
result Simplified models explain neural collapse, emergence, etc.
In this paper, we first give two fundamental principles under a technique to characterize conformal vector fields of (α,β) spaces to be homothetic and determine the local structure of those homothetic fields. Then we use the principles to study conformal vector fields of some classes of (α,β) spaces under certain c…
The first part of this article intends to present the role played by Thom in diffusing Smale's ideas about immersion theory, at a time (1957) where some famous mathematicians were doubtful about them: it is clearly impossible to make the sphere inside out! Around a decade later, M. Gromov transformed Smale's idea in wh…
Large deviation principles for multivariate stochastic volatility models.
problem Understanding the behavior of log-processes in multivariate stochastic volatility models.
method Establishing a comprehensive sample path large deviation principle for log-processes.
result Asymptotic formulas for first exit times and barrier option prices derived from the LDP.
Generalizes Thurston's jiggling lemma for piecewise smooth solutions.
problem Creating piecewise smooth solutions of differential relations without homotopical assumptions.
method Jiggling arbitrary sections of E to construct solutions of R. result Generalization of Thurston's lemma for piecewise smooth solutions of differential relations.
In this paper we are dealing with two classes of mean curvature type problems that generalize the translating soliton problem. A first result proves that the solutions to these problems have unique interior critical points. Using this uniqueness result, we next derive a priori C0 and C1 estimates for the solution…
The paper extends the avoidance principle for mean curvature flows, proving new intersection dimension monotonicity results.
problem Understanding the behavior of intersections in mean curvature flows.
method Proving new intersection dimension monotonicity results for mean curvature flows, Brakke flows, and level set flows.
result The dimension of the intersection of mean curvature flows is non-increasing over time.
Discussing AI's difficulty and physics' simplicity, suggesting AI benefits from physics principles.
problem AI difficulty compared to physics simplicity.
method Drawing on physical intuition and theoretical physics to improve AI.
result AI and physics principles are strongly coupled through sparsity.
A new principle and method improve out-of-distribution detection in generative models.
problem Out-of-distribution detection in deep generative models often fails due to poor likelihood estimates.
method Introducing the Likelihood Path (LPath) principle and new theoretical tools for OOD detection.
result Non-asymptotic provable OOD detection guarantees for variational autoencoders (VAEs).
Lower bounds for eigenvalues on Bakry-Emery manifolds proven.
problem Eigenvalue estimates on Bakry-Emery manifolds.
method Generalised maximum principle and heat kernel estimates.
result Lower bounds for all eigenvalues proven.
Paper presents an action principle for Einstein-Weyl equations in 3D.
problem Finding an action principle for Einstein-Weyl equations.
method Metric affine f(R) gravity action plus additional terms involving Lagrange multipliers and gravitational Chern-Simons contributions.
result The Weyl vector dynamics is governed by a special case of the generalized monopole equation.
One endeavour of modern physical chemistry is to use bottom-up approaches to design materials and drugs with desired properties. Here we introduce an atomistic structure learning algorithm (ASLA) that utilizes a convolutional neural network to build 2D compounds and layered structures atom by atom. The algorithm takes …
Study on curve shortening flow in 3D space curves, showing convexity preservation and avoidance principle.
problem Analyzing the behavior of space curves under curve shortening flow in R3. method Analysis of properties of space curves evolved by the curve shortening flow, including convexity preservation and avoidance principle.
result Orthogonal projections of space curves remain convex, and the Avoidance principle is shown for spherical curves.
This paper improves a local-to-global principle for Morse quasigeodesics.
problem Quantify the size of local neighborhoods for global Morse behavior.
method Estimates in symmetric space to supplement Kapovich-Leeb-Porti's proof.
result Explicit criteria for local-to-global principle verified.
These notes review six lectures given by Prof. Andrea Montanari on the topic of statistical estimation for linear models. The first two lectures cover the principles of signal recovery from linear measurements in terms of minimax risk. Subsequent lectures demonstrate the application of these principles to several pract…
We propose a new active learning by query synthesis approach using Generative Adversarial Networks (GAN). Different from regular active learning, the resulting algorithm adaptively synthesizes training instances for querying to increase learning speed. We generate queries according to the uncertainty principle, but our…
New theorem connects probabilistic permanental point processes to Monge-Ampère equation.
problem Probabilistic interpretation of Monge-Ampère equation boundary value problem.
method Large deviation principles and optimal transport theory.
result Explicit rate function for permanental point processes large deviation.
Dirac structures on tangent bundles provide a unified framework for Lagrange--Dirac dynamical systems.
problem Unified geometric framework for Lagrange--Dirac dynamical systems
method Introducing a Lagrange--Dirac structure on the tangent bundle
result Unified framework for nonholonomic, degenerate Lagrangian, and symmetric systems
Develops an analytic theory for quantum imaginary time evolution.
problem Lack of a first-principle understanding of quantum imaginary time evolution.
method Interprets QITE as a form of VQA trained with QNGD and connects it to the geometric geodesic distance in the quantum Fisher information metric.
result QITE converges faster than vanilla gradient descent-based VQAs, though the advantage is suppressed by Hilbert space dimensionality.
Proposes a new variational principle for Einstein gravity.
problem Formulating Einstein gravity as a gauge theory for the conformal group.
method First order formulation of conformal tractor geometry, variational principle based on abstract principal bundle.
result Provides first order field equations without requiring supplementary constraints.