**Title:** Newton Solver Fails to Converge in Inverse Problem with Burgers' Equation

General
1 


**Description:**
I am trying to solve an inverse problem using FEniCS, where I need to estimate parameters \(A\) and \(C\) in a modified Burgers' equation. The forward model involves solving a nonlinear PDE, but the Newton solver fails to converge during the optimization process. I have tried adjusting solver parameters and reducing the time step, but the issue persists.

**Code:**
```python
from fenics import *
import numpy as np
from scipy.optimize import least_squares

# Parameters
nu = 0.01 / np.pi
T = 1.0
num_steps = 100
dt = T / num_steps

# Create mesh and function space
nx = 255
mesh = IntervalMesh(nx, -1, 1)
V = FunctionSpace(mesh, 'CG', 1)

# Define boundary conditions
u_D = Constant(0.0)
def boundary(x, on_boundary):
    return on_boundary
bc = DirichletBC(V, u_D, boundary)

# Define initial condition
u_n = interpolate(Expression('-sin(pi*x[0])', degree=2), V)

# Define test function and current solution
v = TestFunction(V)
u = Function(V)

# Time loop
def model(p):
    A, C = p
    u_n.assign(interpolate(Expression('-sin(pi*x[0])', degree=2), V))
    model_output = []
    for n in range(num_steps):
        t = n * dt
        F = A * ((u - u_n) / dt) * v * dx + u * u.dx(0) * v * dx + C * inner(grad(u), grad(v)) * dx
        solver_parameters = {
            "nonlinear_solver": "newton",
            "newton_solver": {
                "maximum_iterations": 100,
                "relaxation_parameter": 0.7,
                "relative_tolerance": 1e-8,
                "absolute_tolerance": 1e-10,
            }
        }
        solve(F == 0, u, bc, solver_parameters=solver_parameters)
        model_output.append(u.vector().get_local())
        u_n.assign(u)
    return np.array(model_output)

RuntimeError: 
*** -------------------------------------------------------------------------
*** DOLFIN encountered an error. If you are not able to resolve this issue
*** using the information listed below, you can ask for help at
***
***     https://fenicsproject.discourse.group/
***
*** Remember to include the error message listed below and, if possible,
*** include a *minimal* running example to reproduce the error.
***
*** -------------------------------------------------------------------------
*** Error:   Unable to solve nonlinear system with NewtonSolver.
*** Reason:  Newton solver did not converge because maximum number of iterations reached.
*** Where:   This error was encountered inside NewtonSolver.cpp.
*** Process: 0
*** 
*** DOLFIN version: 2019.2.0.dev0
*** Git changeset:  1c52e837eb54cc34627f90bde254be4aa8a2ae17
*** -------------------------------------------------------------------------