my_ode_solver_matlab_微分方程_神经网络常微分方程_解方程_

nx = 10; dx = 1. / nx; % Wval = {randn(1,nx),randn(nx,1)}; % 可用cell2mat(W(1))转成普通矩阵 load('pdata_10.mat'); Wsav = Wval; lmb = 0.001; loss_val_last = 1e6; times_train = 1; x_space = linspace(0, 1, nx); x_nonsample = 0.05:0.1:0.95; y_space = psy_analytic(x_space); y_nonsample = psy_analytic(x_nonsample); psy_fd = zeros(size(y_space)); psy_fd(1) = 1; %边界 x = sym('x', [1, nx], 'real'); W = {sym('w', [1, nx], 'real'),sym('v', [nx, 1], 'real')}; %W{1}为w，W{2}为v psy_fd_exp = sym('psy_fd_exp', [1, nx], 'real'); % F = loss_function(W,x); % G = jacobian(F,W{1}); % H = gradient(F,W{1}); %G:1*10 H:10*1 for i = 1:times_train F = loss_function(W,x); loss_grad1_exp = jacobian(F,W{1}); loss_grad2_exp = jacobian(F,W{2}); loss_grad1 = double(subs(loss_grad1_exp,[W{1},W{2}',x],[Wval{1},Wval{2}',x_space])); loss_grad2 = double(subs(loss_grad2_exp,[W{1},W{2}',x],[Wval{1},Wval{2}',x_space])); i loss_val = double(subs(F,[W{1},W{2}',x],[Wval{1},Wval{2}',x_space])) lmb if(loss_val>loss_val_last) Wval = Wsav; break else loss_val_last = loss_val; end Wsav = Wval; Wval{1} = Wval{1} - lmb * loss_grad1; Wval{2} = Wval{2} - lmb * loss_grad2'; end for i = 1:nx psy_fd_exp = 1 + x_space(i).*neural_network(W,x_space(i)); psy_fd(i) = double(subs(psy_fd_exp,[W{1},W{2}'],[Wval{1},Wval{2}'])); end figure(1) plot(x_nonsample, y_nonsample, '-ro') hold on load('nndata.mat'); y=sim(net,x_nonsample); plot(x_nonsample, y, '-b*') legend('解析解','拟合解','Location','northwest'); figure(2) plot(x_space, y_space, '-ro') hold on plot(x_space, psy_fd,'-b*') legend('解析解','近似解','Location','northwest'); function f = A(x) f = x + (1 + 3*x.^2)./(1 + x + x.^3); end function f = B(x) f = x.^3 + 2*x + x.^2 .* ((1 + 3*x.^2)./(1 + x + x.^3)); end function f = fun(x, psy) % d(psy)/dx = f(x, psy) f = B(x) - psy.* A(x); end function f = psy_analytic(x) f = (exp((-x.^2)/2)) ./ (1 + x + x.^3) + x.^2; end function f = sigmoid(x) f = 1 ./ (1 + exp(-x)); end function f = sigmoid_grad(x) %1次求导 f = diff(sigmoid(x)); % sigmoid(x).*(1 - sigmoid(x)); end function f = neural_network(W,x) a1 = sigmoid(x*W{1}); f = a1*W{2}; end function f = d_neural_network_dx(W,x) f = W{2}'*W{1}'*sigmoid_grad(x); end function f = loss_function(W,x) loss_sum = 0; for i = 1:10 net_out = neural_network(W, x(i)); psy_t = 1 + x(i).* net_out; d_net_out = d_neural_network_dx(W, x(i)); d_psy_t = net_out + x(i).* d_net_out; func = fun(x(i), psy_t); err_sqr = (d_psy_t - func).^2; loss_sum = loss_sum + err_sqr; end f = loss_sum; end