Kernel-based meshfree collocation method for solving nonlinear and parametric PDEs

Single droplet drying, a fundamental process in spray drying, presents a challenging nonlinear moving boundary diffusion problem. This process is described by a parabolic partial differential equation in a shrinking spherical domain with a Robin mass-transfer boundary condition. Despite extensive recent applications of physics-informed neural networks (PINNs) to solve PDEs, vanilla PINNs often struggle with time-dependent, moving boundary transport problems. This study develops SDD-PINN, a compact and reproducible PINN framework that remains reliable on such evolving domains. To decouple the training from domain shrinkage, we first map the problem to a fixed unit domain via a radius transformation, ξ = r / a ( t ) , yielding a nonlinear advection-diffusion PDE that preserves mass conservation (baseline PINN). Building on this, a compact, physics-motivated PINN recipe is presented comprising: (i) hard enforcement of the initial condition, (ii) a squared transformed radius ( ξ 2 ) as a symmetry-consistent coordinate input to promote regularity near the spherical center, (iii) a logarithmic time reparameterization θ from non-dimensional time τ , (iv) smart collocation sampling, and (v) a combined Adam + L-BFGS optimization schedule. A 2 5 full-factorial experimental design (soft vs. hard initial condition, ξ vs . ξ 2 , τ vs . θ , uniform vs. smart sampling, Adam vs. Adam + L-BFGS) is first conducted on a reference regime to identify a progressive enhancement path. The selected configuration, SDD-PINN (hard initial condition, ξ 2 feature , and θ as input, uniform sampling, and a combined Adam + L-BFGS optimizer), resulted in a mean relative L 2 error 0.021 ± 0.010 against a traditional Crank–Nicolson (CN) reference. Validation across multiple drying conditions spanning shrinkage intensity, diffusivity, and particle size with multi-seed repeats showed that relative to the CN reference, the unified recipe yields a mean relative L 2 error ranging from 2.69×10 −3 to 3.30×10 −2 . These results provide a reproducible, PINN recipe for single droplet drying.

A multi-resolution hybrid Haar wavelet collocation method for solving nonlinear partial differential equations

A novel sixth-order compact numerical algorithm for 3D nonlinear elliptic PDEs on a cubic grid: Relevance to multi-harmonic elliptic BVPs

Heat and mass transfer in radiative magnetohydrodynamics stagnation flow of upper-convected Maxwell fluids over porous sheets with applications to industrial thermal processing

We develop a unified analytical and computational framework that integrates fractal geometry, scaling laws, and artificial intelligence into the study of parabolic quasi-variational inequalities associated with Hamilton-Jacobi-Bellman equations. The analysis is done for irregular domains with boundaries having a Hausdorff dimension [Formula: see text], which enables the influence of the geometric complexity of the fractal interface on the numerical approximation. We develop a generalized overlapping domain decomposition method in a fractal Sobolev space setting. The geometric convergence rate is dictated by a scaling law, with the contraction factor given by [Formula: see text] where [Formula: see text] denotes the overlap width. This result reveals that interface roughness directly controls the decay rate of the iterative error. Additionally, we prove a multifractal maximum norm error estimate of the form [Formula: see text] which shows that the spatial convergence rates depend on the boundary’s fractal dimension. In order to improve the computational efficiency, we have proposed an AI-assisted adaptive overlap method, which relies on the prediction of local spectral radii, and we have shown that it accelerates the convergence significantly. The above results link, in a rigorous way, the fractal geometric structure, the scaling properties of domain decomposition algorithms, and data-driven computational optimization, which provides a novel interdisciplinary tool for the analysis of nonlinear PDEs.

A numerical technique for solving nonlinear fractional PDEs

Abstract The State-Dependent Riccati Equation (SDRE) approach is extensively utilized in nonlinear optimal control as a reliable framework for designing robust feedback control strategies. This work provides an analysis of the SDRE approach, examining its theoretical foundations, error bounds, and numerical approximation techniques. We explore the relationship between SDRE and the Hamilton-Jacobi-Bellman (HJB) equation, deriving residual-based error estimates to quantify its suboptimality. Additionally, we introduce an optimal semilinear decomposition strategy to minimize the residual. From a computational perspective, we analyze two numerical methods for solving the SDRE: the offline–online approach and the Newton–Kleinman iterative method. Their performance is assessed through a numerical experiment involving the control of a nonlinear reaction-diffusion PDE. Results highlight the trade-offs between computational efficiency and accuracy, indicating better performance of the Newton–Kleinman approach in achieving stable and cost-effective solutions in the reported experiments.

Switching event-triggered control of nonlinear parabolic PDE systems via Galerkin/neural-network-based modeling approach

Genuinely multi-dimensional stationarity preserving Finite Volume formulation for nonlinear hyperbolic PDEs

An IMEX scheme for a nonlinear PDE model of tumor angiogenesis

High-order empirical interpolation methods for real-time solution of parametrized nonlinear PDEs

Convergence of Physics-Informed Neural Networks for Fully Nonlinear PDEs

Highly Efficient Symmetry Energy-Preserving SAV Methods for the Nonlinear Hamiltonian PDEs

In this work, we present a complete dynamical analysis of lump, breather, M-shaped, and other waveforms propagating in a nonlinear PDE governing nonlinear low-pass electrical transmission lines. We utilize the Hirota bilinear transformation approach with the help of Mathematica to report a number of wave solutions, including bright and dark lumps, solitons, breathers, and kink waves, along with their periodic and aperiodic forms. Energy distribution, wave interactions, and changes are presented in the form of 3D, contour, and 2D plots, which demonstrate the nonlinear characteristics that govern the dynamics. These results provide a better understanding of the propagation, stability, and interaction of waveforms which are useful in signal and energy transport and also in the construction of complex nonlinear electric circuits.

A new version of the linear integral representation method is developed for solving inverse problems in geophysics. This approach is applied to the interpretation of anomalous time-dependent field data. The reconstruction of field elements is reduced to solving a system of linear algebraic equations (SLAE) with an approximately given right-hand side. Since the matrix elements of this system are derived analytically, the modeling process is significantly simplified. The article also analyzes how the approximation quality of a non-stationary field element depends on the observation network geometry, enabling its optimization for more accurate detection of geological properties. The proposed method for solving inverse problems for hyperbolic partial differential equations with constant coefficients can also be applied to data described by systems of nonlinear PDEs, provided the target field is represented as a composition of components differing in magnitude. Finally, the results of non-stationary gravity field modeling are presented.

Non-intrusive data-driven methodologies, such as POD-DL, offer a powerful approach for constructing surrogate models to address complex parametric problems. POD-DL integrates deep neural networks and follows a multi-step dimensionality reduction process, beginning with a linear reduction via Proper Orthogonal Decomposition (POD) and followed by a nonlinear reduction using a deep autoencoder. A subsequent nonlinear regression, implemented with a forward neural network, accounts for temporal and parametric coefficients. The framework involves numerous hyperparameters, including the number of POD basis modes, network architecture specifications, and typical neural network parameters like learning rate and batch size, all of which influence both training time and model accuracy. In a previous work, we applied this scheme to study gravity currents under the 2D lock-exchange configuration [3] for multiple angles of the initial configuration [4]. The angle of the channel served as a parameter, i.e., different angles generated different dynamics that were learned by the surrogate model. The regression neural network could then predict the dynamics for unseen angles. Now, we extend the methodology to more realistic scenarios, a 3D channel-basin configuration for gravity currents, adapted from [5], analyzing variations in inlet velocities and sediment concentrations. Synthetic data were generated through large-scale parallel finite element simulations, allowing POD-DL to predict deposition maps for unseen parameter values. [1] M. Cracco et al., “Deep learning-based reduced-order methods for fast transient dynamics”, Arxiv Preprint 2212.07737, 2022. [2] S. Fresca and A. Manzoni, “POD-DL-ROM: Enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition”, Comput. Methods Appl. Mech. Engrg., 2022. [3] V. K. Birman, B. A. Battandier, E. Meiburg, and P. F. Linden, “Lock-exchange flows in sloping channels”, Journal of Fluid Mechanics, 577:53–77, 2007. [4] R. M. Velho, A. M. Cortes, G. F. Barros, F. A. Rochinha, and A. L. G. A. Coutinho, Advances in Data-Driven Reduced Order Models Using Two-Stage Dimension Reduction for Coupled Viscous Flow and Transport. Finite Elements in Analysis and Design, vol. 248, 2025. [5] T. Spychala, J. T. Eggenhuisen, M. Tilston and F. Pohl,The influence of basin setting and turbidity current properties on the dimensions of submarine lobe elements, SEDIMENTOLOGY, 2020.

Local Rigidity of Quasi–Lie Brackets on Quaternionic Banach Modules and Applications to Nonlinear PDEs

Fully Nonlinear Elliptic PDEs in Thin Domains with Oblique Boundary Condition

This study analyzes the impact of local thermal non-equilibrium on the bioconvection flow of hybrid nanofluid across a slender extending sheet containing gyrotactic bacteria using artificial neural networks trained using a Bayesian regularization backpropagation approach (ANN-BRS). The effects of magnetic fields, thermal radiation, and Hall current are all things related to fluid flow. The suggested model has particular applicability in microscale drug delivery systems, where gyrotactic microorganisms and hybrid nanofluid can be employed to control nutrition and medication dispersion under non-equilibrium temperature circumstances. It can be used in lab-on-chip and organ-on-chip technologies to improve bio-mixing and accurate heat control. The model also applies to micro-solar collectors and porous micro-heat exchangers, which use hybrid nanoparticles to boost thermal efficiency. It can also be used in bioreactors and biomedical cooling systems, where local thermal non-equilibrium effects and ANN-based prediction allow for precise control of heat, mass, and microbe transfer, resulting in optimal performance. Similarity transformations are used to convert the original nonlinear PDEs into non-dimensional ODEs and the bvp4c program is applied to numerically resolve the resulting boundary-value problem. The training, testing, and validation processes yield the expected outcomes for every scenario based on the chosen data points. Regression analysis, histograms of error, and mean square error (MSE) metrics are employed to assess the ANN-BRS model's outcome. The liquid phase heat thermal profile increases as the interphase heat transfer parameter values rise, while the solid phase thermal profile decreases.Graphical abstract