Identification of partial differential equations from noisy data with integrated knowledge discovery and embedding using evolutionary neural networks

Identification of underlying partial differential equations(PDEs)for complex systems remains a formidable challenge.In the present study,a robust PDE identification method is proposed,demonstrating the ability to extract accurate governing equations under noisy conditions without prior knowledge.Specifically,the proposed method combines gene expression programming,one type of evolutionary algorithm capable of generating unseen terms based solely on basic operators and functional terms,with symbolic regression neural networks.These networks are designed to represent explicit functional expressions and optimize them with data gradients.In particular,the specifically designed neural networks can be easily transformed to physical constraints for the training data,embedding the discovered PDEs to further optimize the metadata used for iterative PDE identification.The proposed method has been tested in four canonical PDE cases,validating its effectiveness without preliminary information and confirming its suitability for practical applications across various noise levels.

Hydration mechanisms of smithsonite from DFT-D calculations and MD simulations

Investigation on the mineralwater interactions is crucial for understanding the subsequent interfacial reactions.Currently,the hydration mechanisms of smithsonite are still obscure.In this paper,the adsorption of H_(2)O at different coverage rates on smithsonite(101)surface was innovatively investigated using density-functional theory(DFT)calculations and molecular dynamics(MD)simulations by analyzing adsorption model,interaction energy,atomic distance,density of state,electron density difference,concentration profile,radial distribution function and self-diffusion coefficient.We found that single H_(2)O preferred to be dissociated on smithsonite(101)surface via the interaction of surface Zn with the Ow of H_(2)O and H-bond between Hw of H_(2)O and surface Os.However,dissociation adsorption and molecular adsorption coexisted on the smithsonite surface at a high coverage rate of H_(2)O,and dissociation adsorption remained the main adsorption mechanism.Moreover,we found the interaction between smithsonite surface and H_(2)O was weakened as a function of H_(2)O coverage,which was because the presence of interlayer H_(2)O and different layers of H_(2)O decreased the reactivity of the smithsonite surface.The H_(2)O is mainly adsorbed on the smithsonite surface by forming three layers of H_(2)O(about 10–15Å),with the ordering degree gradually decreasing.

Insights into the adsorption performance and mechanism of hydrated Ca ion on talc(001) basal surface from DFT calculation

The utilization of Ca ion as assistant depressant of CMC on talc has been widely reported.Thus,the study on the adsorption mechanism of Ca ion on talc surface is very crucial for understanding the performance of CMC on talc depression.In this paper,mechanism insights into hydrated Ca ion adsorption on talc(001) basal surface were creatively provided using DFT calculation.[Ca(H_(2)O)_6]^(2+) and [Ca(OH)(H_(2)O)_(3)]^(+) were determined as the effective hydrate components for Ca ion adsorption,and the top O site was the most favorable position for their adsorptions on talc surface.Furthermore,the adsorption mechanisms of [Ca(H_(2)O)_6]^(2+) and [Ca(OH)(H_(2)O)_(3)]^(+) on talc surface were found to be not the Ca-O chemical bond,but the hydrogen bonding formed by the H atom of the H_(2)O ligand and the surface O atom.H_(2)O acted like a bridge to connect them to the talc surface.Moreover,the hydrogen bonding was formed due to the hybridization of H 1s orbital with the O 2s,O 2p orbitals.Simultaneously,electrons transferred between the H atom and the surface O atom.This work provides theoretical insights into the Ca ion adsorption on talc surface,which can help deeply understand the talc flotation using CMC as depression.