Physics-informed neural networks have emerged as a prominent new method for solving differential equations. While conceptually straightforward, they often suffer training difficulties that lead to relatively large discretization errors or the failure to obtain correct solutions. In this paper we introduce invariant physics-informed neural networks for ordinary differential equations that admit a finite-dimensional group of Lie point symmetries. Using the method of equivariant moving frames, a differential equation is invariantized to obtain a, generally, simpler equation in the space of differential invariants. A solution to the invariantized equation is then mapped back to a solution of the original differential equation by solving the reconstruction equations for the left moving frame. The invariantized differential equation together with the reconstruction equations are solved using a physics-informed neural network, and form what we call an invariant physics-informed neural network. We illustrate the method with several examples, all of which considerably outperform standard non-invariant physics-informed neural networks.

Designing and implementation of an intrusion detection system in any database environment has emerged as an absolute necessity in the recent years. Detection of both, the outsider attack and privilege abuse from within the organisation, has become a fundamental need for maintenance of dynamic, scalable and reinforced databases. Proposed advanced approach, malicious query detection using fuzzy and cluster analysis (MQDFCA) operates in a seamless manner and efficaciously performs detection and prevention of transactions that are intrusive in nature, within a database environment, thus shielding the vital data stored in a database from any unauthorised/malicious access or modifications. The method utilises concepts of machine learning like fuzzy logic, association rule mining and clustering algorithms at various stages to validate a newly generated transaction at role segment, profile segment and the rule validation segment. The degree of adherence of user supplied queries within a transaction to the previously generated user roles, transaction profiles and extracted rules is used to categorise the transaction as non-malicious or malicious. The efficaciousness of proposed methodology in detection of intrusions is exemplified from the results of the experiments conducted on the synthetic dataset yielding recall and precision values of 93% and 98% respectively.