Physics-Informed Neural Networks (PINNs) have emerged as a key tool in Scientific Machine Learning since their introduction in 2017, enabling the efficient solution of ordinary and partial differential equations using sparse measurements. Over the past few years, significant advancements have been made in the training and optimization of PINNs, covering aspects such as network architectures, adaptive refinement, domain decomposition, and the use of adaptive weights and activation functions. A notable recent development is the Physics-Informed Kolmogorov-Arnold Networks (PIKANS), which leverage a representation model originally proposed by Kolmogorov in 1957, offering a promising alternative to traditional PINNs. In this review, we provide a comprehensive overview of the latest advancements in PINNs, focusing on improvements in network design, feature expansion, optimization techniques, uncertainty quantification, and theoretical insights. We also survey key applications across a range of fields, including biomedicine, fluid and solid mechanics, geophysics, dynamical systems, heat transfer, chemical engineering, and beyond. Finally, we review computational frameworks and software tools developed by both academia and industry to support PINN research and applications. 

From PINNs to PIKANs: Recent Advances in Physics-Informed Machine
  Learning

This paper is devoted to the study of an optimal control problem for a generalized multi-group reaction–diffusion SIR epidemic model, with heterogeneous nonlinear incidence rates. The proposed model incorporates a wide range of spatiotemporal epidemic models. Primarily, the ones used to describe the propagation of zoonotic and sexually transmitted diseases, as well as epidemics that propagate disparately within populations. For the aforementioned diseases, dividing the susceptible, infected and recovered populations into several subpopulations is necessary in order to capture all the possible ways of the disease transmission. This makes the problem of finding the possible optimal control strategies and the division of the available control resources complicated. To address this problem mathematically, for each subpopulation, we introduce two types of control variables, namely vaccination for the susceptible and treatment for the infected. The existence and uniqueness of a biologically feasible solution to the proposed model, for fixed controls, is derived by means of a truncation technique and a semigroup approach. Moreover, first-order necessary optimality conditions for the introduced optimal control problem are obtained using the adjoint state method. Finally, numerical simulations are performed for a two-group epidemic model with particular incidence rates and by considering three cases in the maximal control resources allowed for each subpopulation. 

https://link.springer.com/content/pdf/10.1007/s40435-022-01030-3.pdf

Optimal control for a multi-group reaction–diffusion SIR model with heterogeneous incidence rates

The aim of this work is to describe the dynamics of a discrete fractional-order reaction–diffusion FitzHugh–Nagumo model. We established acceptable requirements for the local asymptotic stability of the system’s unique equilibrium. Moreover, we employed a Lyapunov functional to show that the constant equilibrium solution is globally asymptotically stable. Furthermore, numerical simulations are shown to clarify and exemplify the theoretical results. 

The FitzHugh-Nagumo Model Described by Fractional Difference Equations: Stability and Numerical Simulation

Oncolytic viral immunotherapy is gaining considerable prominence in the realm of chronic diseases treatment and rehabilitation. Oncolytic viral therapy is an intriguing therapeutic approach due to its low toxicity and dual function of immune stimulations. This work aims to design a soft computing approach using stupendous knacks of neural networks (NNs) optimized with Bayesian regularization (BR), i.e. NNs-BR, procedure. The constructed NNs-BR technique is exploited in order to determine the approximate numerical treatment of the nonlinear multi-delayed tumor virotherapy (TVT) models in terms of the dynamic interactions between the tumor cells free of viruses, tumor cells infected by viruses, viruses, and cytotoxic T-lymphocytes (CTLs). The strength of state-of-the-art numerical approach is incorporated to develop the reference dataset for the variation in the infection rate for tumor cells, virus-free tumor cell clearance rate by CTLs, CTLs clearance rate for infectious tumor cells, the natural lifecycle of infectious tumor cells, the natural lifecycle of viral cell, the natural lifecycle of CTLs cells, tumor cells free of viruses’ maximum proliferation rate, production of tumor cells with an infection, CTLs simulated ratio for infectious tumor cells, CTLs simulated ratios for virus-free cells and delay in time. The dataset is randomly chosen/segmented for training-testing-validation samples to construct the NNs models optimized with backpropagated BR representing the approximate numerical solutions of the dynamic interactions in the TVT model. The performance of the designed NNs-BR technique is accessed/evaluated and outcomes are found in good agreement with the reference solutions having the range of accuracy from 10[Formula: see text] to 10[Formula: see text]. The efficacy of NNs-BR paradigm is further substantiated after rigorous analysis on regression metrics, learning curves on MSE, and error histograms for the dynamics of TVT model. 

Novel intelligent Bayesian computing networks for predictive solutions of nonlinear multi-delayed tumor oncolytic virotherapy systems

This paper considers the dynamical properties of a space and time discrete fractional reaction–diffusion epidemic model, introducing a novel generalized incidence rate. The linear stability of the equilibrium solutions of the considered discrete fractional reaction–diffusion model has been carried out, and a global asymptotic stability analysis has been undertaken. We conducted a global stability analysis using a specialized Lyapunov function that captures the system’s historical data, distinguishing it from the integer-order version. This approach significantly advanced our comprehension of the complex stability properties within discrete fractional reaction–diffusion epidemic models. To substantiate the theoretical underpinnings, this paper is accompanied by numerical examples. These examples serve a dual purpose: not only do they validate the theoretical findings, but they also provide illustrations of the practical implications of the proposed discrete fractional system. 

On Stability of a Fractional Discrete Reaction–Diffusion Epidemic Model

This paper investigates the global dynamics for a class of multigroup SIR epidemic model with time fractional-order derivatives and reaction–diffusion. The fractional order considered in this paper is in (0; 1], which the propagation speed of this process is slower than Brownian motion leading to anomalous subdiffusion. Furthermore, the generalized incidence function is considered so that the data itself can flexibly determine the functional form of incidence rates in practice. Firstly, the existence, nonnegativity, and ultimate boundedness of the solution for the proposed system are studied. Moreover, the basic reproduction number R0 is calculated and shown as a threshold: the disease-free equilibrium point of the proposed system is globally asymptotically stable when R0 ≤ 1, while when R0 > 1, the proposed system is uniformly persistent, and the endemic equilibrium point is globally asymptotically stable. Finally, the theoretical results are verified by numerical simulation.

/pdf/global-dynamics-for-a-class-of-reaction-diffusion-multigroup-gs15gyzt.pdf

Global dynamics for a class of reaction–diffusion multigroup SIR epidemic models with time fractional-order derivatives

Physics informed neural networks (PINNs) are a type of deep learning framework which can use the knowledge of physical laws to facilitate the learning of various forward and inverse problems. PINNs have displayed significant efficiency in numerical computation, but there are still some limitations such as their inability in tackling differential equations with oscillations or singular perturbation. To alleviate this difficulty, a new deep learning framework called improved fractional physics informed neural networks (IFPINNs) is proposed in this paper. Compared to PINNs, the calculation in the weight propagation between layers is replaced with nonlinear functions rather than linear ones. This new architecture allows for a better usage of physical information of differential equations, thus leading to an improved fitting quality for the cases with high oscillations or singular perturbation. Under certain conditions, IFPINNs can be degenerated into PINNs. Moreover, this paper investigates the performance of IFPINNs in dealing with fractional differential equations with time-fractional orders ranging from 1 to 2. The numerical results demonstrate the proposed architecture can achieve a higher accuracy for various integral and fractional differential equations. 

A class of improved fractional physics informed neural networks

In this paper, the numerical method for a multiterm time-fractional reaction–diffusion equation with classical Robin boundary conditions is considered. The full discrete scheme is constructed with the L1-finite difference method, which entails using the L1 scheme on graded meshes for the temporal discretisation of each Caputo fractional derivative and using the finite difference method on uniform meshes for spatial discretisation. By dealing with the discretisation of Robin boundary conditions carefully, sharp error analysis at each time level is proven. Additionally, numerical results that can confirm the sharpness of the error estimates are presented.

/pdf/local-error-estimate-of-an-l1-finite-difference-scheme-for-13b3lvbi.pdf

Local Error Estimate of an L1-Finite Difference Scheme for the Multiterm Two-Dimensional Time-Fractional Reaction–Diffusion Equation with Robin Boundary Conditions

Global dynamics for a class of discrete fractional epidemic model with reaction-diffusion

In recent years, the epidemic model with anomalous diffusion has gained popularity in the literature. However, when introducing anomalous diffusion into epidemic models, they frequently lack physical explanation, in contrast to the traditional reaction–diffusion epidemic models. The point of this paper is to guarantee that anomalous diffusion systems on infectious disease spreading remain physically reasonable. Specifically, based on the continuous-time random walk (CTRW), starting from two stochastic processes of the waiting time and the step length, time-fractional space-fractional diffusion, time-fractional reaction–diffusion and fractional-order diffusion can all be naturally introduced into the SIR (S: susceptible, I: infectious and R: recovered) epidemic models, respectively. The three models mentioned above can also be applied to create SIR epidemic models with generalized distributed time delays. Distributed time delay systems can also be reduced to existing models, such as the standard SIR model, the fractional infectivity model and others, within the proper bounds. Meanwhile, as an application of the above stochastic modeling method, the physical meaning of anomalous diffusion is also considered by taking the SEIR (E: exposed) epidemic model as an example. Similar methods can be used to build other types of epidemic models, including SIVRS (V: vaccine), SIQRS (Q: quarantined) and others. Finally, this paper describes the transmission of infectious disease in space using the real data of COVID-19.

A class of anomalous diffusion epidemic models based on CTRW and distributed delay

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