We consider the problem - Δ p ⁢ u + a ⁢ ( x ) ⁢ | u | p - 2 ⁢ u = | u | q - 2 ⁢ u {-\Delta_{p}u+a(x)\lvert u\rvert^{p-2}u=\lvert u\rvert^{q-2}u} in ℝ N {\mathbb{R}^{N}} , where 1 < p < N {1<p<N} , p < q < p * = N ⁢ p N - p {p<q<p^{*}=\frac{Np}{N-p}} , Δ p {\Delta_{p}} is the p -Laplacian operator, and the potential function a is positive, bounded and verifies suitable decay assumptions. The existence of infinitely many solutions of the equation is proved via the truncation method.

Physics-informed neural networks have emerged as a transformative framework for addressing operator learning and inverse problems involving the Korteweg–de Vries (KdV) equation for internal solitary waves. By integrating physical constraints with data-driven optimization, PINNs overcome the critical challenges of parameter unmeasurability in the KdV equation for internal solitary waves in two-layer fluid systems. This work addresses two problems: (1) operator learning constructs a mapping from parameters to solutions, enabling wave evolution predictions from unknown parameters. Comparative studies demonstrate prediction errors as low as 10−4 when using 1000 training points. (2) Inverse problem solving leverages sparse and potentially noisy observational data with physics-regularized constraints to invert nonlinear coefficients successfully. Compared to conventional approaches, this end-to-end differentiable paradigm unifies operator learning and inverse problem-solving while overcoming mesh discretization errors and high-dimensional parameter space iteration costs. The method shows effectiveness for internal wave problems in stratified fluids, providing both accurate forward modeling and robust parameter inversion capabilities, even under noise.

X-ray absorption spectroscopy (XAS) is a critical analytical technique for comprehensively characterizing the electronic configurations and atomic structures of materials. The rapid growth in both data volume and complexity, driven by modern synchrotron radiation facilities, necessitates computational frameworks capable of efficiently processing large-scale XAS datasets. To address this need, we introduce XASDAML , a machine-learning-based platform that integrates the entire data processing workflow. The framework coordinates key operational processes, including spectral–structural descriptor generation, predictive modeling and performance validation, while facilitating statistical analyses through principal component decomposition and clustering algorithms to uncover latent patterns within datasets. Designed with modular architecture, the system enables independent modification or enhancement of individual components, ensuring flexibility to meet evolving analytical demands. Implemented through a Jupyter Notebook -based interface, the platform ensures accessibility for researchers. The framework is validated with two case studies: (i) copper-foil EXAFS data show that it can predict coordination numbers and radial distribution functions; and (ii) XANES spectra of the spin-crossover complex Fe(phen) ₃ uncover bond-length changes between the low-spin and high-spin states. Comprehensive validation highlights robust toolkit functionalities, including statistical descriptor analyses, spectral visualization, and prediction of widely employed structural descriptors closely reflecting local atomic environments. By establishing standardized and extensible procedures for integrating machine learning into XAS analysis, XASDAML enhances research efficiency, promotes richer data insights, and provides a versatile computational resource tailored to the expanding needs of XAS research.