More Like this

1. Travelling Waves for Adaptive Grid Discretizations of Reaction Diffusion Systems III: Nonlinear Theory

   https://doi.org/10.1007/s10884-022-10143-4  

   Hupkes, H. J.; Van Vleck, E. S. (April 2022, Journal of Dynamics and Differential Equations)

   Abstract In this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE and establish the existence of travelling waves. In particular, we consider the time dependent spatial mesh adaptation method that aims to equidistribute the arclength of the solution under consideration. We assume that this equidistribution is strictly enforced, which leads to the non-local problem with infinite range interactions that we derived in Hupkes and Van Vleck (J Dyn Differ Eqn, 2021). Using the Fredholm theory developed in Hupkes and Van Vleck (J Dyn Differ Eqn, 2021) we setup a fixed point procedure that enables the travelling PDE waves to be lifted to our spatially discrete setting. 

   more » « less
2. Travelling Waves for Adaptive Grid Discretizations of Reaction Diffusion Systems I: Well-Posedness

   https://doi.org/10.1007/s10884-021-10013-5  

   Hupkes, H. J.; Van Vleck, E. S. (June 2022, Journal of Dynamics and Differential Equations)

   Abstract In this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE. In particular, we consider a commonly used time dependent moving mesh method that aims to equidistribute the arclength of the solution under consideration. We assume that the discrete analogue of this equidistribution is strictly enforced, which allows us to reduce the effective dynamics to a scalar non-local problem with infinite range interactions. We show that this reduced problem is well-posed and obtain useful estimates on the resulting nonlinearities. In the sequel papers (Hupkes and Van Vleck in Travelling waves for adaptive grid discretizations of reaction diffusion systems II: linear theory; Travelling waves for adaptive grid discretizations of reaction diffusion systems III: nonlinear theory) we use these estimates to show that travelling waves persist under these adaptive spatial discretizations. 

   more » « less
3. Using asymptotic embedding methods for dynamic estimation of spatial fields with mobile sensors

   https://doi.org/10.1109/CDC.2018.8619751  

   Demetriou, Michael A. (December 2018, 2018 IEEE Conference on Decision and Control (CDC))

   This paper proposes an asymptotic embedding method for the dynamic reconstruction of spatially varying fields. By assuming that the spatial field is the solution to an elliptic partial differential equation, then the elliptic PDE is embedded into a parabolic PDE which represents the time-varying estimator. An important advantage of the dynamic estimation scheme is the significant reduction in the use of sensing devices needed to reconstruct the spatial field. Static estimation schemes impose stringent conditions on the regularity of a regression matrix, which links the basis functions to the number of measurements. To further improve the performance of the dynamic estimator, a guidance scheme is proposed that repositions mobile sensors within the spatial field, which is linked to the performance of the dynamic estimator. Extensions to collaborative estimation and optimization of the placement of static sensors are also summarized to provide an integrated account on all facets of optimal dynamic estimation of spatial fields. Numerical simulations for spatial fields in one and two spatial dimensions are included along with a comparison of static reconstruction as quantified by the number of sensing devices required and the relative error. 

   more » « less
4. Nonlocal Attention Operator: Materializing Hidden Knowledge Towards Interpretable Physics Discovery

   Yu, Yue; Liu, Ning; Lu, Fei; Gao, Tian; Jafarzadeh, Siavash; Silling, Stewart (December 2024, 38th Conference on Neural Information Processing Systems (NeurIPS 2024))

   Despite the recent popularity of attention-based neural architectures in core AI fields like natural language processing (NLP) and computer vision (CV), their potential in modeling complex physical systems remains underexplored. Learning problems in physical systems are often characterized as discovering operators that map between function spaces based on a few instances of function pairs. This task frequently presents a severely ill-posed PDE inverse problem. In this work, we propose a novel neural operator architecture based on the attention mechanism, which we refer to as the Nonlocal Attention Operator (NAO), and explore its capability in developing a foundation physical model. In particular, we show that the attention mechanism is equivalent to a double integral operator that enables nonlocal interactions among spatial tokens, with a data-dependent kernel characterizing the inverse mapping from data to the hidden parameter field of the underlying operator. As such, the attention mechanism extracts global prior information from training data generated by multiple systems, and suggests the exploratory space in the form of a nonlinear kernel map. Consequently, NAO can address ill-posedness and rank deficiency in inverse PDE problems by encoding regularization and achieving generalizability. We empirically demonstrate the advantages of NAO over baseline neural models in terms of generalizability to unseen data resolutions and system states. Our work not only suggests a novel neural operator architecture for learning interpretable foundation models of physical systems, but also offers a new perspective towards understanding the attention mechanism. Our code and data accompanying this paper are available at https://github.com/fishmoon1234/NAO. 

   more » « less
5. Invariant-Domain Preserving High-Order Time Stepping: II. IMEX Schemes

   https://doi.org/10.1137/22M1505025  

   Ern, Alexandre; Guermond, Jean-Luc (October 2023, SIAM Journal on Scientific Computing)

   We consider high-order discretizations of a Cauchy problem where the evolution operator comprises a hyperbolic part and a parabolic part with diffusion and stiff relaxation terms. We propose a technique that makes every implicit-explicit (IMEX) time stepping scheme invariant-domain preserving and mass conservative. Following the ideas introduced in Part I on explicit Runge--Kutta schemes, the IMEX scheme is written in incremental form. At each stage, we first combine a low-order and a high-order hyperbolic update using a limiting operator, then we combine a low-order and a high-order parabolic update using another limiting operator. The proposed technique, which is agnostic to the space discretization, allows one to optimize the time step restrictions induced by the hyperbolic substep. To illustrate the proposed methodology, we derive four novel IMEX methods with optimal efficiency. All the implicit schemes are singly diagonal. One of them is A-stable and the other three are L-stable. The novel IMEX schemes are evaluated numerically on systems of stiff ordinary differential equations and nonlinear conservation equations. 

   more » « less