Domain decomposition subspace neural network method for solving linear and nonlinear partial differential equations

Bone regeneration is a complex biological process, and the repair of critical-sized bone defects is often difficult, highlighting the need to develop biodegradable scaffolds that can effectively synergize with tissue repair. This paper introduces a model-based bone repair design framework that matches the kinetics of material degradation with the spatiotemporal stages of bone repair. We systematically evaluated the degradation mechanisms and tunability of four classes of biomaterials-metals (Mg, Fe, Zn), ceramics (HA, β-TCP, bioactive glass), polymers (PLGA, PCL), and composites-and mapped established mathematical models (linear partial differential equations, sigmoid curves, response-diffusion partial differential equations, and mechanobiological finite element models) to specific bone defect sizes and clinical scenarios. Based on this classification, we propose a four-step clinical design to guide researchers in completing the following steps: (1) calculating patient-specific healing timelines using first-order heuristics; (2) selecting appropriate material classes based on mechanical and biological requirements; (3) designing degradation kinetics to match the healing stages; and (4) validating scaffold-tissue alignment. To achieve quantitative assessment, we introduce the synchronization index (SI), a provisional index designed to quantify the correlation between actual implant degradation and patient-specific target curves. While the SI and the healing timeline formula are currently proposed only as conceptual design tools and require prospective experimental and clinical validation, their integration provides a structured path for scaffold development, moving it from experimental trial and error to precise, model-based engineering design. The goal of this framework is to facilitate the rational design of smart, resorbable implants that dynamically match the biological and mechanical requirements of bone regeneration.

In this study, an efficient method is presented for the analysis of the Klein-Gordon (KG) and Sine-Gordon (SG) equations with initial value problems. KG and SG equations are hyperbolic partial differential equations that possess the capability to model phenomena in both quantum and classical mechanics, as well as solitons and condensed matter physics. KG equation represents a relativistic wave equation while SG equation represents the d’Alembert operator with a nonlinear sine term of the dependent variable. The proposed method is based on applying the coupling of Aboodh transformation and Adomian decomposition method (ADM) to partial differential equations and this study is limited to KG and SG equations. The non-linear term is replaced by Adomian polynomials for the index n. The elements of the dependent variable are substituted within the recurrence relation by their respective Aboodh transform components corresponding to the same index. Consequently, the nonlinear problem is addressed in a direct manner, devoid of any linearization or discretization processes. Illustrations are presented to demonstrate the efficacy and veracity of the method. A comparison of the findings with the precise solution indicates that the method proved to be efficient because the results are in closed agreement with the exact solution (errors = 0 with just 5–6 terms). The study concludes that this method can be applied to a variety of linear and nonlinear partial differential equation because Aboodh Adomian Decomposition Method (AADM) provides accurate numerical solutions for linear and nonlinear problems, and can be extended to solve other problems arising in applied science.

PDE-GANet: Partial differential equation discovery powered by adversarial learning.

We introduce finite-element Gaussian processes (FEGPs), a novel physics-informed machine learning approach for solving inverse problems involving steady-state, linear partial differential equations (PDEs). Our framework combines a Gaussian process prior for the unknown solution function with a likelihood that incorporates the PDE in its weak form, using a finite-element approximation. This approach offers significantly better scalability than physics-informed Gaussian processes (PIGPs), which rely on the strong form of the PDE. Through numerical experiments on a range of synthetic benchmark problems, we show that FEGPs offer results which outperform PIGPs, and are competitive with physics-informed neural networks (PINNs) with improved uncertainty quantification.

In this paper, we investigate the properties of the boundedness of fractional integral operators Kα defined on general measure metric spaces. We study their action in Lebesgue spaces Lp(Y), Morrey spaces Lφp(Y), and extend our analysis to fractional Sobolev spaces Wα,p(Y). Using classical dyadic decomposition and the Hardy–Littlewood maximal operator, we establish sharp bounds for Kα in terms of kernel parameters and the geometric structure of the space. A significant contribution of this work is the proof that Kα is bounded from Wα,p(Y) to Lq(Y), where thus linking our operator-theoretic framework with the theory of nonlocal and fractional partial differential equations. These results provide valuable tools for studying regularity, a priori estimates, and solution mappings in nonlocal problems involving the fractional Laplacian and related operators on irregular or non- Euclidean domains.

This paper proposes a disturbance observer-based spatiotemporal event-triggered anti-disturbance control method for Markov jump linear parameter-varying partial differential equation (LPV-PDE) systems subject to disturbances. Based on the Takagi–Sugeno (T–S) fuzzy model, the original nonlinear system is approximated as a fuzzy LPV-PDE system. To reduce the number of sensors, pointwise measurements are introduced, based on which a spatiotemporal event-triggered mechanism is employed to alleviate the communication burden. Furthermore, a disturbance observer is developed to estimate the unknown disturbances modeled by an exogenous system. A fuzzy anti-disturbance point controller is then designed by combining the observer output with traditional state feedback. Sufficient conditions for the fuzzy LPV-PDE system to be exponentially stable are given by constructing a Lyapunov–Krasovskii functional (LKF). Finally, a numerical example is presented to demonstrate the effectiveness of the proposed strategy.

The variational data assimilation problem provides a standard approach to compute the unknown initial value for prediction of natural phénomena. We use an ill-posed optimal control problem for the determination of the initial state at the time of the first available measurements. In this paper, an efficient algorithmic schema to approximate the initial condition for linear evolution equations of reaction-diffusion and convection type is presented. In this paper, we propose a decision-support tool for identifying the initial condition of an inverse problem governed by a linear parabolic partial differential equation (PDE) modeling pollution concentration in a bounded domain . More precisely, the objective is to reconstruct , where denotes the final observation time, throughout the entire domain , using only the available data from a subdomain . This reconstruction is based on a non-standard approach to null-controllability.

Abstract We propose an explicit quantum framework for numerically simulating general linear partial differential equations (PDEs), extending previous work (Guseynov et al 2025 Phys. Rev. Res. 7 033100) to incorporate (a) Robin boundary conditions—which include Neumann and Dirichlet conditions as special cases–(b) inhomogeneous terms, and (c) variable coefficients in space and time. Our approach begins with a general finite-difference discretization and applies the Schrödingerisation technique to transform the resulting system into one that admits unitary quantum evolution, enabling quantum simulation. For the Schrödinger equation corresponding to the discretized PDE, we construct an efficient block-encoding of the Hamiltonian H that scales polylogarithmically with the number of grid points N . This encoding is compatible with quantum signal processing and allows for the implementation of the evolution operator e − i H t . The explicit circuit construction in our method permits complexity to be measured in fundamental gate units–namely, CNOT gates and single-qubit rotations–bypassing the inefficiencies of oracle queries. Consequently, the overall algorithm scales polynomially with N and linearly with the spatial dimension d . Under certain input/output assumptions our method achieves a polynomial speedup in N and an exponential advantage in d for a wide class of PDEs, thereby mitigating the classical curse of dimensionality. The validity and efficiency of the proposed approach are further substantiated by numerical simulations. By explicitly defining the quantum operations and quantifying their resource requirements, our approach offers a practical alternative for numerically solving PDEs, distinct from others that rely on oracle queries and purely asymptotic scaling methods.

ABSTRACT We investigate the transition between stability and chaos in the damped Klein‐Gordon equation, a fundamental model for wave propagation and energy dissipation. Using semigroup methods and spectral criteria, we derive explicit thresholds that determine when the system exhibits asymptotic stability and when it displays strong chaotic dynamics, including Devaney and distributional chaos as well as topological mixing. The results yield a classification of the dynamical regimes in terms of damping, stiffness, and propagation parameters, showing that the system admits only two long‐term behaviours: Convergence to equilibrium or chaos. This dichotomy not only unifies and extends previous partial results but also highlights the mechanisms by which linear partial differential equations can generate complex dynamics typically associated with nonlinear systems. Potential applications arise in acoustics, wave mechanics, and signal transmission, where predicting the onset of chaos versus stability is of practical importance.

A physics student's first encounter with the method of separation of variables for solving linear partial differential equations can be confusing if the starting assumption—that the solution can be written as a product of functions, each of which depends on only one of the independent variables—is put forward without motivation. We describe a way of introducing the method that provides such motivation by presenting it first as an application of the general idea of expanding an unknown function in a basis, then showing the conventional approach as a way to streamline the calculation. The presentation is designed for students who have not yet studied function spaces, orthogonality, and the like.

The Laplace residual power series technique was previously proposed for solving linear and nonlinear fractional differential equations in the sense of Caputo using independent theories and related expansions. In this research, we focus on modifying this technique to more effectively solve linear and nonlinear fractional partial differential equations within the framework of the conformable fractional derivative. This modification was employed to construct wave soliton solutions for a class of conformable-fractional partial differential equations. Supported by proven theories, new expansions are used to enhance the modified technique. We test the proposed method with four different types of nonlinear time-conformable-fractional partial differential equations. The solutions are shown graphically for various orders of the fractional derivative. We compare the results with the exact solutions in cases of the classical derivative. All results demonstrate the method's simplicity, efficiency, and accuracy.

The boundary layer flow due to a surface stretching with a power law distribution in the presence of a transverse magnetic field is studied. A approximate Numerical solution for the flow problem has been obtained by solving the governing equations using Numerical Technique. A magnetic field is applied transversely to the direction of the flow. Adopting the similarity transformation, governing non linear partial differential equation of the problem are transformed to non linear ordinary differential equations. Then the numerical solution of the problem is derived using Quasilinearization method, for different values of the dimensionless parameter. The results obtained show that the flow field is influenced appreciably by the presence of chemical reaction and magnetic field.

Dynamic system theory of a nonlinear partial differential equation

We present a quantum solver for partial differential equations based on a flexible matrix product operator representation. Utilizing midcircuit measurements and a state-dependent norm correction, this scheme overcomes the restriction of unitary operators. Hence, it allows for the direct implementation of a broad class of differential equations governing the dynamics of classical and quantum systems. The capabilities of the framework are demonstrated for linear and nonlinear partial differential equations using the example of the linearized Euler equations with absorbing boundaries and the nonlinear Burgers’ equation. For a turbulence data set, we demonstrate potential advantages of the quantum-tensor scheme over its classical counterparts.

This paper studies a quantum simulation technique for solving the Fokker-Planck equation. Traditional semi-discretization methods often fail to preserve the underlying Hamiltonian dynamics and may even modify the Hamiltonian structure, particularly when incorporating boundary conditions. We address this challenge by employing the Schrödingerization method – it converts any linear partial and ordinary differential equation with non-Hermitian dynamics into systems of Schrödinger-type equations. It does so via the so-called warped phase transformation that maps the equation into one higher dimension. We explore the application in two distinct forms of the Fokker-Planck equation. For the conservation form, we show that the semidiscretization-based Schrödingerization is preferable, especially when dealing with non-periodic boundary conditions. Additionally, we analyze the Schrödingerization approach for unstable systems that possess positive eigenvalues in the real part of the coefficient matrix or differential operator. Our analysis reveals that the direct use of Schrödingerization has the same effect as a stabilization procedure. For the heat equation form, we propose a quantum simulation procedure based on the time-splitting technique, and give explicitly its corresponding quantum circuit. We discuss the relationship between operator splitting in the Schrödingerization method and its application directly to the original problem, illustrating how the Schrödingerization method accurately reproduces the time-splitting solutions at each step. Furthermore, we explore finite difference discretizations of the heat equation form using shift operators. Utilizing Fourier bases, we diagonalize the shift operators, enabling efficient simulation in the frequency space. Providing additional guidance on implementing the diagonal unitary operators, we conduct a comparative analysis between diagonalizations in the Bell and the Fourier bases, and show that the former generally exhibits greater efficiency than the latter.

Abstract We present an algorithm for training predictive surrogate models using a loss function that combines two sources of similar data. The method jointly optimises surrogate model weights and a hyperparameter that controls the relative influence of each source, encouraging convergence toward mappings that represent their respective signals best. A differentiable loss function leverages gradient‐based optimisation to achieve this task. The resulting model can achieve higher quality results when unknown errors distort the signal from either source. Using the proposed methodology, experimental case studies are conducted to train a surrogate model to produce representations of the simple diffusion equation, the nonlinear Schrödinger equation, and the circular rotating shallow‐water equations. Thus, surrogates trained with the ratio‐coupled loss are shown to approximate linear and nonlinear partial differential equations for any number of prediction variables and under varying noise conditions.

In this paper, the method of fundamental solutions (MFS) is first developed for solving direct problems in bi-layer materials in the biomedical field of optical fluorescence. The governing system of second-order linear partial differential equations (PDEs) for the emission and excitation fluences is transformed into a single fourth-order PDE with appropriate boundary and interface matching conditions. The MFS is subsequently further developed, in conjunction with a constrained minimization regularization procedure, to solve nonlinear inverse optical fluorescence tomography problems. Numerical results confirm the accuracy, stability and versatility of the proposed meshless technique.

High-resolution imaging provides dense trajectories of migrating cells, flocking animals, and synthetic active particles, from which interaction laws can be determined with a wide variety of methods. Yet, distinguishing whether front-back or lateral biases seen in such data reflect intrinsic anisotropy in the interaction kernel or emergent correlations that are nevertheless produced by isotropic pairwise interaction forces remains an open challenge. We resolve this ambiguity by deriving a linear partial differential equation that connects measurable two-point velocity correlations to an unknown, distance- and angle-dependent interaction kernel. Turing-like instabilities can occur which allows for dipolar or quadrupolar patterns to arise even when agents interact according to an underlying attraction-repulsion law that is angle-independent. We then show that incorporating a weak velocity-alignment force can interfere with anisotropic pattern formation by suppressing dipolar patterns. We validate these predictions with agent-based simulations and provide design guidance for experiments that seek to discriminate intrinsic anisotropy from emergent effects.

In this note, we shall study a new, short proof of the general theorem of Necas about the solvability of linear partial differential equations in the Banach space setting. The complexity of this proof does not seem to be greater than that of the Lax-Milgram theorem, and since the theorem of Necas is strictly stronger than the Lax-Milgram theorem, the author hopes that his new proof will help the theorem of Necas to gain prominence in PDE and functional analysis lectures.