it is technically possible to use PSO for predicting the output, although not by directly applying PSO. The regression problem can be formulated as a least-squares problem, and an objective function can be constructed from it, which can then be minimized using PSO. Here is an example, but please note that it can be somewhat tedious, as MATLAB's particleswarm() is designed for single-objective optimization.
%% Data
x = 0:5; % input vector
y = [2.1 7.7 13.6 27.2 40.9 61.1]; % output vector
%% Data processing
n = length(x);
Sx = sum(x);
Sx2 = sum(x.^2);
Sx3 = sum(x.^3);
Sx4 = sum(x.^4);
Sy = sum(y);
Sxy = sum(x.*y);
Sx2y = sum((x.^2).*y);
%% Least-square Regression model: lsy(x) = p1·x² + p2·x + p3;
% Sx2*p1 + Sx*p2 + n*p3 = Sy ... Eq.(1)
% Sx3*p1 + Sx2*p2 + Sx*p3 = Sxy ... Eq.(2)
% Sx4*p1 + Sx3*p2 + Sx2*p3 = Sx2y ... Eq.(3)
% p3 = (Sy - (Sx2*p1 + Sx*p2))/n ... from Eq.(1)
% Sx3*p1 + Sx2*p2 + Sx*((Sy - (Sx2*p1 + Sx*p2))/n) = Sxy ... Eq.(4)
% Sx4*p1 + Sx3*p2 + Sx2*((Sy - (Sx2*p1 + Sx*p2))/n) = Sx2y ... Eq.(5)
% p2 = (Sxy - Sx*Sy/n - Sx3*p1 + Sx*Sx2/n*p1)/(Sx2 - Sx*Sx/n) ... from Eq.(4)
% Sx4*p1 + Sx3*((Sxy - Sx*Sy/n - Sx3*p1 + Sx*Sx2/n*p1)/(Sx2 - Sx*Sx/n)) + Sx2*((Sy - (Sx2*p1 + Sx*((Sxy - Sx*Sy/n - Sx3*p1 + Sx*Sx2/n*p1)/(Sx2 - Sx*Sx/n))))/n) - Sx2y = 0 ... Eq.(6)
%% Make Eq.(6) a convex function so that PSO can be used
fun = @(p1) (Sx4*p1 + Sx3*((Sxy - Sx*Sy/n - Sx3*p1 + Sx*Sx2/n*p1)/(Sx2 - Sx*Sx/n)) + Sx2*((Sy - (Sx2*p1 + Sx*((Sxy - Sx*Sy/n - Sx3*p1 + Sx*Sx2/n*p1)/(Sx2 - Sx*Sx/n))))/n) - Sx2y).^2;
nvar = 1;
p1 = particleswarm(fun, nvar)
p2 = (Sxy - Sx*Sy/n - Sx3*p1 + Sx*Sx2/n*p1)/(Sx2 - Sx*Sx/n)
p3 = (Sy - (Sx2*p1 + Sx*p2))/n
%% Find the coefficient of determination, R²
xbar = mean(x);
ybar = mean(y);
dev = y - ybar;
Sdev = sum(dev.^2);
lsy = @(x) p1*x.^2 + p2*x + p3;
err = y - lsy(x);
Serr = sum(err.^2);
Rsq = (Sdev - Serr)/Sdev % R-square
%% Plot result
xx = 0:0.01:5;
plot(x, y, 'o', 'markersize', 12, 'linewidth', 2), hold on
plot(xx, lsy(xx)), grid on
xlabel('x'), ylabel('y')
title('Polynomial Regression using PSO')
