  An improved physics-informed neural network (IPINN) algorithm with four
output functions and four physics constraints, which possesses neuron-wise
locally adaptive activation function and slope recovery term, is appropriately
proposed to obtain the data-driven vector localized waves, including vector
solitons, breathers and rogue waves (RWs) for the Manakov system with initial
and boundary conditions, as well as data-driven parameters discovery for
Manakov system with unknown parameters. The data-driven vector RWs which also
contain interaction waves of RWs and bright-dark solitons, interaction waves of
RWs and breathers, as well as RWs evolved from bright-dark solitons are learned
to verify the capability of the IPINN algorithm in training complex localized
wave. In the process of parameter discovery, routine IPINN can not accurately
train unknown parameters whether using clean data or noisy data. Thus we
introduce parameter regularization strategy with adjustable weight coefficients
into IPINN to effectively and accurately train prediction parameters, then find
that once setting the appropriate weight coefficients, the training effect is
better as using noisy data. Numerical results show that IPINN with parameter
regularization shows superior noise immunity in parameters discovery problem.
