More Like this

1. Efficient quantum algorithm for dissipative nonlinear differential equations

   https://doi.org/10.1073/pnas.2026805118  

   Liu, Jin-Peng; Kolden, Herman Øie; Krovi, Hari K.; Loureiro, Nuno F.; Trivisa, Konstantina; Childs, Andrew M. (August 2021, Proceedings of the National Academy of Sciences)

   null (Ed.)

   Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic n-dimensional ordinary differential equations. Assuming R < 1 , where R is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity T 2 q   poly ( log ⁡ T , log ⁡ n , log ⁡ 1 / ϵ ) / ϵ , where T is the evolution time, ϵ is the allowed error, and q measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for R ≥ 2 . Finally, we discuss potential applications, showing that the R < 1 condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of R. 

   more » « less
2. FINITE DIMENSIONAL MODELS FOR RANDOM MICROSTRUCTURES

   https://doi.org/10.1090/tpms/1168  

   M. Grigoriu (May 2022, Theory of probability and mathematical statistics)

   Finite dimensional (FD) models, i.e., deterministic functions of space depending on finite sets of random variables, are used extensively in applications to generate samples of random fields Z(x) and construct approximations of solutions U(x) of ordinary or partial differential equations whose random coefficients depend on Z(x). FD models of Z(x) and U(x) constitute surrogates of these random fields which target various properties, e.g., mean/correlation functions or sample properties. We establish conditions under which samples of FD models can be used as substitutes for samples of Z(x) and U(x) for two types of random fields Z(x) and a simple stochastic equation. Some of these conditions are illustrated by numerical examples. 

   more » « less
3. Needles in a haystack. An algorithmic approach to the classification of 4d $$ \mathcal{N} $$ = 2 SCFTs

   https://doi.org/10.1007/JHEP03(2022)210  

   Kaidi, Justin; Martone, Mario; Rastelli, Leonardo; Weaver, Mitch (March 2022, Journal of High Energy Physics)

   A bstract There is a well-known map from 4d $$ \mathcal{N} $$ N = 2 superconformal field theories (SCFTs) to 2d vertex operator algebras (VOAs). The 4d Schur index corresponds to the VOA vacuum character, and must be a solution with integral coefficients of a modular differential equation. This suggests a classification program for 4d $$ \mathcal{N} $$ N = 2 SCFTs that starts with modular differential equations and proceeds by imposing all known constraints that follow from the 4d → 2d map. This program becomes fully algorithmic once one specifies the order of the modular differential equation and the rank (complex dimension of the Coulomb branch) of the $$ \mathcal{N} $$ N = 2 theory. As a proof of concept, we apply the algorithm to the study of rank-two $$ \mathcal{N} $$ N = 2 SCFTs whose Schur indices satisfy a fourth-order untwisted modular differential equation. Scanning over a large number of putative cases, only 15 satisfy all of the constraints imposed by our algorithm, six of which correspond to known 4d SCFTs. More sophisticated constraints can be used to argue against the existence of the remaining nine cases. Altogether, this indicates that our knowledge of such rank-two SCFTs is surprisingly complete. 

   more » « less
4. Universal differential equations for glacier ice flow modelling

   https://doi.org/10.5194/gmd-16-6671-2023  

   Bolibar, Jordi; Sapienza, Facundo; Maussion, Fabien; Lguensat, Redouane; Wouters, Bert; Pérez, Fernando (January 2023, Geoscientific Model Development)

   Abstract. Geoscientific models are facing increasing challenges to exploit growing datasets coming from remote sensing. Universal differential equations (UDEs), aided by differentiable programming, provide a new scientific modelling paradigm enabling both complex functional inversions to potentially discover new physical laws and data assimilation from heterogeneous and sparse observations. We demonstrate an application of UDEs as a proof of concept to learn the creep component of ice flow, i.e. a nonlinear diffusivity differential equation, of a glacier evolution model. By combining a mechanistic model based on a two-dimensional shallow-ice approximation partial differential equation with an embedded neural network, i.e. a UDE, we can learn parts of an equation as nonlinear functions that then can be translated into mathematical expressions. We implemented this modelling framework as ODINN.jl, a package in the Julia programming language, providing high performance, source-to-source automatic differentiation (AD) and seamless integration with tools and global datasets from the Open Global Glacier Model in Python. We demonstrate this concept for 17 different glaciers around the world, for which we successfully recover a prescribed artificial law describing ice creep variability by solving ∼ 500 000 ordinary differential equations in parallel. Furthermore, we investigate which are the best tools in the scientific machine learning ecosystem in Julia to differentiate and optimize large nonlinear diffusivity UDEs. This study represents a proof of concept for a new modelling framework aiming at discovering empirical laws for large-scale glacier processes, such as the variability in ice creep and basal sliding for ice flow, and new hybrid surface mass balance models. 

   more » « less
5. Boundary criticality of the O(N) model in d = 3 critically revisited

   https://doi.org/10.21468/SciPostPhys.12.4.131  

   Metlitski, Max (January 2022, SciPost Physics)

   It is known that the classical O(N) O ( N ) model in dimension d > 3 d gt; 3 at its bulk critical point admits three boundary universality classes:the ordinary, the extra-ordinary and the special. For the ordinarytransition the bulk and the boundary order simultaneously; theextra-ordinary fixed point corresponds to the bulk transition occurringin the presence of an ordered boundary, while the special fixed pointcorresponds to a boundary phase transition between the ordinary and theextra-ordinary classes. While the ordinary fixed point survives in d = 3 d = 3 ,it is less clear what happens to the extra-ordinary and special fixedpoints when d = 3 d = 3 and N \ge 2 N ≥ 2 .Here we show that formally treating N N as a continuous parameter, there exists a critical value N_c > 2 N c gt; 2 separating two distinct regimes. For 2 \leq N < N_c 2 ≤ N < N c the extra-ordinary fixed point survives in d = 3 d = 3 ,albeit in a modified form: the long-range boundary order is lost,instead, the order parameter correlation function decays as a power of \log r log r .For N > N_c N gt; N c there is no fixed point with order parameter correlations decayingslower than power law. We discuss several scenarios for the evolution ofthe phase diagram past N = N_c N = N c .Our findings appear to be consistent with recent Monte Carlo studies ofclassical models with N = 2 N = 2 and N = 3 N = 3 .We also compare our results to numerical studies of boundary criticalityin 2+1D quantum spin models. 

   more » « less