This convergence of high‐performance computing and MSM has transformed parallel algorithms (see Table 2), which are the engines of multiphysics modeling. Multiple scale analysis has a number of specialized topics, including the analysis of singular perturbation problems, the application to partial differential equations, and emerging areas of research. Can we harness biological learning to design more efficient algorithms and architectures? Artificial intelligence through deep learning is an exciting recent development that has seen remarkable success in solving problems, which are difficult for humans. Typical examples include chess and Go, as well as the classical problem of image recognition, that, although superficially easy, engages broad areas of the brain.
Efficient prediction of static and dynamical responses of functional graded beams using sparse multiscale patches
However, the PatchTST model7 proved the efficiency of Transformer models in time series analysis, which exploits patching and channel independence to model time series data, pinpointing that Transformer architecture is still a powerful model with some adaptation and architectural adjustments. Following PatchTST, other Transformer models have been developed for time series and proved high capability in dealing with Multi-scale analysis high-dimensional time series10. Triformer25 designs a patch attention with linear complexity and variable specific parameters to enhance accuracy.
PARALLEL AND HIGH‐PERFORMANCE COMPUTING
In traffic prediction often capturing spatial-temporal dependencies at multiple scales is required. To address the mentioned requirement, MT-STNets is designed in30, for prediction of both fine-grained traffic conditions on individual roads and coarse-grained traffic flows across urban areas. More recently, Pathformer8 is proposed which exploits adaptive pathways to capture multi-scale temporal relations in an adaptive manner by automatically selecting patches of different resolutions, which uses separate set of parameters for each temporal granularity in its design. Where machine learning reveals correlation, multiscale modeling can probe whether the correlation is causal; where multiscale modeling identifies mechanisms, machine learning, coupled with Bayesian methods, can quantify uncertainty. This natural synergy presents exciting challenges and new opportunities in the biological, biomedical, and behavioral sciences.28 On a more fundamental level, there is a pressing need to develop the appropriate theories to integrate machine learning and multiscale modeling. Can we use prior physics-based knowledge to avoid overfitting or non-physical predictions?
Machine learning seeks to infer the dynamics of biological, biomedical, and behavioral systems
Where $\epsilon$ is a small parameter, $x$ is the independent variable, and $f_n(x)$ are the coefficients of the expansion. Modelingadvanced materials accurately is extremely complex because of the high numberof variables at play. The materials in question are heterogeneous in Software quality assurance nature,meaning they have more than one pure constituent, e.g. carbon fiber + polymerresin or sedimentary rock + gaseous pores.
Lighthill introduced a more general version in 1949.Later Krylov and Bogoliubov and Kevorkian and Cole introduced thetwo-scale expansion, which is now the more standard approach. One technique used to account for microstructural nuances is to use an analytical equation to model behavior. Then, they generate a relationship between all relevant variables that match the observed outcomes. Alternatively, modern approaches derive these sorts of models using coordinate transforms, like in the method of normal forms,3 as described next.
MULTISCALE MODELING 2.0
- The general approach used in all these methods is to average out relatively insignificant microscopic details in order to obtain reasonable computational efficiency while preserving the essential microscopic‐level details.
- While the considerations above and the motivation to combine MSM and ML can benefit several disciplines, it is particularly relevant for chemical, biomolecular, and biological engineers.
- The lack of sufficient data is a common problem in modeling biological, biomedical, and behavioral systems.
- Quantum MC techniques provide a direct and potentially efficient means for solving the many‐body Schrödinger equation of QM.58 The simplest quantum MC technique, variational MC, is based on a direct application of MC integration to calculate multidimensional integrals of expectation values such as the total energy.
The key is that the user must be very aware of the assumptions and bounds of their model when employing one of these techniques. Unsupervised learning aims at drawing inferences from datasets consisting of input data without labeled responses. The most common types of unsupervised learning techniques include clustering and density estimation used for exploratory data analysis to identify hidden patterns or groupings. I acknowledge insightful discussions from students of BE559 MSM of chemical and biological systems at the University of Pennsylvania, where the topics in this review are discussed as a one‐semester course.
- HPC has remained a sustained and powerful driving force for multiphysics modeling and scientific computing and central to applications in science, engineering, and medicine.
- I would also like to thank Keith E. Gubbins for introducing me to MSM over two decades ago.
- The materials in question are heterogeneous in nature,meaning they have more than one pure constituent, e.g. carbon fiber + polymerresin or sedimentary rock + gaseous pores.
- In traffic prediction often capturing spatial-temporal dependencies at multiple scales is required.
- As evidenced by the table, each of the multi-scale embedding, channel-wise encoder and multi-step decoder modules contribute to performance promotion.
- On behalf of all authors, the corresponding author states that there is no conflict of interest.
We begin by outlining a summary of historical developments of governing equations and foundations for multiphysics modeling in Table 1. Machine learning and multiscale modeling interact on the parameter level via constraining parameter spaces, identifying parameter values, and analyzing sensitivity and on the system level via exploiting the underlying physics, constraining design spaces, and identifying system dynamics. Machine learning provides the appropriate tools towards supplementing training data, preventing overfitting, managing ill-posed problems, creating surrogate models, and quantifying uncertainty with the ultimate goal being to explore massive design spaces and identify correlations. Multiscale modeling integrates the underlying physics towards identifying relevant features, exploring their interaction, elucidating mechanisms, bridging scales, and understanding the emergence of function with the ultimate goal of predicting system dynamics and identifying causality. We are riding the wave of a paradigm shift in the development of MSM methods due to rapid development and changes in HPC infrastructure (see Figure 2) and advances in ML methods.