Relatively recently, as part of my research work, I had to numerically solve a system of differential equations, and instead of using ready-made programs written in Fortran, I decided to write my own in Python. As one of the intermediate steps, I encountered a system of differential equations, which you can see in the attachment. I found a module that solves differential equations using machine learning, and it seemed to me that it would be suitable for solving this particular problem. But a question arose. How do I specify all the boundary conditions? The ones that are set straightforwardly as specific numbers are easy to specify, but what should I do with those where the relationship between the derivative of the function and the function itself is specified?

Attachment:
SDE

My code:

import matplotlib.pyplot as plt

from neurodiffeq import diff
from neurodiffeq.solvers import Solver1D, Solver2D
from neurodiffeq.conditions import IVP, DirichletBVP2D, DirichletBVP, BundleIVP, DoubleEndedBVP1D
from neurodiffeq.networks import FCNN, SinActv


#PARAMS
N = 500

EPS = 1e-5

V0 = 50.0
NU = 1e-4
DELTA_V = 10.0
ZA = 1.0
ALPHA = 1.0

CA0 = 1e-2

P = 1.0

def F(y):
    return y*DELTA_V


def ode_system(Cplus, Cminus, Ca, y):
    return [V0*diff(Cplus,y)+diff(Cplus*DELTA_V + diff(Cplus,y),y),
            V0*diff(Cminus,y) + diff(-Cminus*DELTA_V + diff(Cminus,y),y),
            ALPHA*V0*diff(Ca,y) + diff(-ZA*Ca*DELTA_V + diff(Ca,y),y),
            -(Cminus+ZA*Ca-Cplus),
            ]


conditions = [
    DirichletBVP(
        t_0=0, u_0=P,
        t_1=1, u_1=1.0,
    ),
    IVP(t_0=1.0, u_0=(1.0-ZA*CA0)),
    IVP(t_0=1.0, u_0=CA0),
]

nets = [FCNN(actv=SinActv), FCNN(actv=SinActv), FCNN(actv=SinActv) , FCNN(actv=SinActv)]

solver = Solver1D(ode_system, conditions, t_min=0.0, t_max=1.0, nets=nets)
solver.fit(max_epochs=N)
solution = solver.get_solution()

t = np.linspace(0.0, 1.0, N)
Cplus, Cminus, Ca = solution(t, to_numpy=True)

fig, axs = plt.subplots(1, 2)

axs[0].plot(t, Cplus, label="C+")
axs[0].plot(t, Cminus, label="C-")
axs[0].plot(t, Ca, label="Ca")

axs[1].plot(t, F(t), label='F')

axs[0].legend()
axs[1].legend()
plt.show()