7. Federated worker setup

7. Federated worker setup#

Parameter estimation times depend one a series of factors, e.g. the dimensionality of the estimation problem or the solution landscape. In most cases, the model simulation to evaluate a candidate solution vector \(\theta\) is the time-limiting step, which is further stretched with increased numbers of experimental replicates. To address this issue, estim8 introduces as setup employing federated computing principles for differentiable cost functions, effectively yielding a second parallelization layer for the simulation of model replicates.

Essentially, federated Workers are launched as gRPC services bound to a unique host and port network address. Evaluation calls for replicate specific parameter sets in the objective function are then distributed to these workers using PyTensor and pytensor_federated.

federated_workers

Using a federated worker setup, heavy computations can be effectively distributed among machines, exploiting parralel computer architectures like workstations or computer. clusters.

Imports#

import sys 
sys.path.append(r'../../')

from estim8.models import FmuModel
from estim8.error_models import BaseErrorModel, LinearErrorModel
from estim8 import visualization, datatypes, Estimator, utils
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns

# load and init model
SimpleBatchModel = FmuModel(path=r'SimpleBatch.fmu')
# import datasheet
data = pd.read_excel(r'SimpleBatch_Data.xlsx', index_col=0)

We start by creating an artificial noisy dataset of 3 replicates like in Notebook 4. Modeling experimental replicates.

# define number of replicates and IDs
replicate_IDs = [str(i) for i in range(1,4)]

# create a parameter mapping
replicate_specific_parameters = ["X0"]
parameter_mapping = utils.ModelHelpers.ParameterMapping(
    mappings=[utils.ModelHelpers.ParameterMapper(
        global_name = param,
        replicate_ID = r_ID,
        value = 0.1 + np.random.random()/10  # use a random turbulence
    )
    for r_ID in replicate_IDs
    for param in replicate_specific_parameters],
    default_parameters = SimpleBatchModel.parameters,
    replicate_IDs = replicate_IDs
)


# create 2 artificial datasets
def make_experiment(parameters, model: FmuModel, original_data: pd.DataFrame, error_model: BaseErrorModel = LinearErrorModel(), rID: str = datatypes.Constants.single_id, resample=4):
    simulation = model.simulate(t0=0, t_end=10, stepsize=0.1, parameters=parameters)

    measurements = []
    for obs in original_data.columns:
        model_prediction = simulation[obs]
        # get timepoints of measurements from datesheet above
        timepoints = original_data[obs].dropna().index
        values = model_prediction.interpolate(timepoints)
        # get noisy values by resampling the data according to erro models distribution
        for _ in range(resample):
            values = error_model.get_sampling(values, 1)[0]

        measurements.append(
            datatypes.Measurement(
                name = model_prediction.name,
                timepoints=timepoints,
                values=values,
                error_model=error_model,
                replicate_ID=rID
            )
        )
    return datatypes.Experiment(measurements, replicate_ID=rID)


# create noisy data
artificial_data = dict()
for r_ID in replicate_IDs:
    replicate_parameters = parameter_mapping.replicate_handling(
        replicate_ID=r_ID
    )

    artificial_data[r_ID] = make_experiment(parameters=replicate_parameters, original_data=data, model=SimpleBatchModel, error_model=LinearErrorModel(offset=0.05), rID=r_ID)

# plot data
fig, axes = plt.subplots(
    ncols=2,
    nrows=1,
    figsize=(15, 5)
)

for (r_ID, experiment), color in zip(artificial_data.items(),sns.color_palette("colorblind") ):
    for ax, measurement in zip(axes, experiment.measurements):
        visualization.plot_measurement(ax=ax, measurement=measurement, color=color, ecolor=color)
        ax.set_xlabel('Time [h]')
        ax.legend()
../_images/20b7bfe3a2bffd6d75914db61b6161878b8b04de18e18027acc320d25045b634.png

7.1 Federated workers on a single machine#

To use this setup on a single local machine, pass the keyword argument federated_workersto the Estimator class methods.

Parameter estimation#

# define unknowns and bounds
bounds = {f"X0_{r_ID}": [0.01, 0.5] for r_ID in replicate_IDs}
bounds["mu_max"] = [0.1, 0.8] 


# instantiate Estimator
estimator = Estimator(
    model=SimpleBatchModel,
    bounds=bounds,
    data=artificial_data,
    metric="WSS",
    parameter_mapping=parameter_mapping,
    t=[0, 10, 0.1]
)


estimates, info = estimator.estimate(
    method="de",
    max_iter=1000,
    optimizer_kwargs={"disp": True, "updating": "deferred"},
    n_jobs=1,
    federated_workers=4                                         # use 4 federated workers for parallel simulation
)

_ = visualization.plot_estimates(estimates, estimator, only_measured=True)

Launching 4 workers...
testing workers..
2024-11-06 15:24:37,920 [INFO] Client 1949739579664-15268-624 connected to localhost:9500
2024-11-06 15:24:37,946 [INFO] Closing evaluation stream
2024-11-06 15:24:37,948 [INFO] Client 1949739829568-15268-624 connected to localhost:9501
2024-11-06 15:24:37,970 [INFO] Closing evaluation stream
2024-11-06 15:24:37,972 [INFO] Client 1949739833168-15268-624 connected to localhost:9502
2024-11-06 15:24:37,994 [INFO] Closing evaluation stream
2024-11-06 15:24:37,996 [INFO] Client 1949739840272-15268-624 connected to localhost:9503
2024-11-06 15:24:38,018 [INFO] Closing evaluation stream
Worker test succesfull.
2024-11-06 15:24:38,634 [INFO] Client 1949739996960-15268-624 connected to localhost:9503
2024-11-06 15:24:39,192 [INFO] Client 1949739996240-15268-624 connected to localhost:9500
2024-11-06 15:24:39,846 [INFO] Client 1949739997680-15268-624 connected to localhost:9502
differential_evolution step 1: f(x)= 26869.17550546418
differential_evolution step 2: f(x)= 9394.542480717359
differential_evolution step 3: f(x)= 9394.542480717359
differential_evolution step 4: f(x)= 1690.9599493810802
differential_evolution step 5: f(x)= 1690.9599493810802
differential_evolution step 6: f(x)= 1690.9599493810802
differential_evolution step 7: f(x)= 1690.9599493810802
differential_evolution step 8: f(x)= 1690.9599493810802
differential_evolution step 9: f(x)= 1690.9599493810802
differential_evolution step 10: f(x)= 1690.9599493810802
differential_evolution step 11: f(x)= 1690.9599493810802
differential_evolution step 12: f(x)= 1690.9599493810802
differential_evolution step 13: f(x)= 1690.9599493810802
differential_evolution step 14: f(x)= 1690.9599493810802
differential_evolution step 15: f(x)= 1657.7098625153628
differential_evolution step 16: f(x)= 1657.7098625153628
differential_evolution step 17: f(x)= 1519.877707181494
differential_evolution step 18: f(x)= 1519.877707181494
differential_evolution step 19: f(x)= 1519.877707181494
differential_evolution step 20: f(x)= 1519.877707181494
differential_evolution step 21: f(x)= 1409.2591297655406
differential_evolution step 22: f(x)= 1409.2591297655406
differential_evolution step 23: f(x)= 1409.2591297655406
differential_evolution step 24: f(x)= 1380.6225538906763
differential_evolution step 25: f(x)= 1380.6225538906763
differential_evolution step 26: f(x)= 1375.4915887689017
differential_evolution step 27: f(x)= 1375.4915887689017
differential_evolution step 28: f(x)= 1375.4915887689017
differential_evolution step 29: f(x)= 1375.4915887689017
differential_evolution step 30: f(x)= 1375.4915887689017
differential_evolution step 31: f(x)= 1372.2566468163504
differential_evolution step 32: f(x)= 1360.8349341981907
differential_evolution step 33: f(x)= 1358.871190463822
differential_evolution step 34: f(x)= 1351.6071020689199
differential_evolution step 35: f(x)= 1351.6071020689199
differential_evolution step 36: f(x)= 1351.6071020689199
differential_evolution step 37: f(x)= 1351.6071020689199
differential_evolution step 38: f(x)= 1351.6071020689199
differential_evolution step 39: f(x)= 1350.7356462071089
differential_evolution step 40: f(x)= 1350.7356462071089
differential_evolution step 41: f(x)= 1348.5973302143843
differential_evolution step 42: f(x)= 1346.8961419432055
differential_evolution step 43: f(x)= 1346.8961419432055
differential_evolution step 44: f(x)= 1346.8961419432055
Polishing solution with 'L-BFGS-B'
Shutting down workers...
../_images/309c94751cb04e7eb547562291f4f81385c4a3236beb7513fb1fa7e9bd5e58ab.png ../_images/4fd416ffb241e499e88be84e7e124db2352cb9b3a31290da369f2b3445e3f705.png ../_images/29950fb1442973aa5fe89ccd2cbd97da9f8ad9d6cff2f778a0265bbea5d6aaa7.png

Uncertainty quantification methods#

To scale up the procesces, similar to the estimate method above, you can define a number of federated_workers.

# monte carlo sampling
mc_samples = estimator.mc_sampling(
    method='de',
    max_iter=1000,
    n_jobs=1,
    mcs_at_once=4,
    federated_workers=16,
    n_samples=100
)

Launching 16 workers...
testing workers..
2024-11-06 15:27:27,190 [INFO] Client 1949745292352-15268-624 connected to localhost:9500
2024-11-06 15:27:27,232 [INFO] Closing evaluation stream
2024-11-06 15:27:27,235 [INFO] Client 1949745292352-15268-624 connected to localhost:9501
2024-11-06 15:27:27,260 [INFO] Closing evaluation stream
2024-11-06 15:27:27,263 [INFO] Client 1949745292352-15268-624 connected to localhost:9502
2024-11-06 15:27:27,285 [INFO] Closing evaluation stream
2024-11-06 15:27:27,288 [INFO] Client 1949745292352-15268-624 connected to localhost:9503
2024-11-06 15:27:27,312 [INFO] Closing evaluation stream
2024-11-06 15:27:27,314 [INFO] Client 1949745292352-15268-624 connected to localhost:9504
2024-11-06 15:27:27,337 [INFO] Closing evaluation stream
2024-11-06 15:27:27,339 [INFO] Client 1949745292352-15268-624 connected to localhost:9505
2024-11-06 15:27:27,363 [INFO] Closing evaluation stream
2024-11-06 15:27:27,365 [INFO] Client 1949745292352-15268-624 connected to localhost:9506
2024-11-06 15:27:27,389 [INFO] Closing evaluation stream
2024-11-06 15:27:27,391 [INFO] Client 1949745292352-15268-624 connected to localhost:9507
2024-11-06 15:27:27,421 [INFO] Closing evaluation stream
2024-11-06 15:27:27,423 [INFO] Client 1949745292352-15268-624 connected to localhost:9508
2024-11-06 15:27:27,446 [INFO] Closing evaluation stream
2024-11-06 15:27:27,449 [INFO] Client 1949745292352-15268-624 connected to localhost:9509
2024-11-06 15:27:27,470 [INFO] Closing evaluation stream
2024-11-06 15:27:27,472 [INFO] Client 1949745292352-15268-624 connected to localhost:9510
2024-11-06 15:27:27,495 [INFO] Closing evaluation stream
2024-11-06 15:27:27,498 [INFO] Client 1949745292352-15268-624 connected to localhost:9511
2024-11-06 15:27:27,521 [INFO] Closing evaluation stream
2024-11-06 15:27:27,524 [INFO] Client 1949745292352-15268-624 connected to localhost:9512
2024-11-06 15:27:27,546 [INFO] Closing evaluation stream
2024-11-06 15:27:27,549 [INFO] Client 1949745292352-15268-624 connected to localhost:9513
2024-11-06 15:27:27,571 [INFO] Closing evaluation stream
2024-11-06 15:27:27,574 [INFO] Client 1949745292352-15268-624 connected to localhost:9514
2024-11-06 15:27:27,597 [INFO] Closing evaluation stream
2024-11-06 15:27:27,600 [INFO] Client 1949745292352-15268-624 connected to localhost:9515
2024-11-06 15:27:27,637 [INFO] Closing evaluation stream
Worker test succesfull.
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Shutting down workers...

7.2 Distributing workloads among different machines#

Employing this federated setup, one can launch workers on one or more machines and just pass the network adresses to the Estimator object that runs the optimization task. Assume on both machines the model and data from above are already defined:

Parameter estimation#

# code run on machine A: worker nodes
print(" \n Machine A:")
estimator_A = Estimator(
    model = SimpleBatchModel,
    data=data,
    bounds = {"X0": [0.01, 0.5], "mu_max": [0.2, 0.8]},
    t=[0,10, 0.1]
)

estimator_A.launch_workers(
    host= "localhost",              # use the real host adress of machine A her
    ports=list(range(9500,9506))    # port adresses for workers
)


print("-------------------------------------------------------")


# code run on machine B: parameter estimation
print(" \n Machine B:")
estimator_B = Estimator(
    model = SimpleBatchModel,
    data=data,
    bounds = {"X0": [0.01, 0.5], "mu_max": [0.2, 0.8]},
    t=[0,10, 0.1]
)

hosts_and_ports = [("localhost", port) for port in list(range(9500,9506))]

# set the hosts_and_ports attribute 
estimator_B.hosts_and_ports = hosts_and_ports

estimator_B.estimate(method='de', max_iter=1000, n_jobs=2, federated_workers=True)

 
 Machine A:
Launching 6 workers...

c:\Users\Latour\projects\estim8-s-test\examples\..\estim8\estimator.py:243: UserWarning: n_workers argument overriden by ports.
  warn(f"n_workers argument overriden by ports.")

-------------------------------------------------------
 
 Machine B:
testing workers..
2024-11-06 15:46:14,759 [INFO] Client 1949752452976-15268-624 connected to localhost:9500
2024-11-06 15:46:14,794 [INFO] Closing evaluation stream
2024-11-06 15:46:14,797 [INFO] Client 1949752458448-15268-624 connected to localhost:9501
2024-11-06 15:46:14,819 [INFO] Closing evaluation stream
2024-11-06 15:46:14,822 [INFO] Client 1949757921344-15268-624 connected to localhost:9502
2024-11-06 15:46:14,843 [INFO] Closing evaluation stream
2024-11-06 15:46:14,846 [INFO] Client 1949757924176-15268-624 connected to localhost:9503
2024-11-06 15:46:14,872 [INFO] Closing evaluation stream
2024-11-06 15:46:14,874 [INFO] Client 1949752452496-15268-624 connected to localhost:9504
2024-11-06 15:46:14,901 [INFO] Closing evaluation stream
2024-11-06 15:46:14,903 [INFO] Client 1949752452832-15268-624 connected to localhost:9505
2024-11-06 15:46:14,927 [INFO] Closing evaluation stream
Worker test succesfull.

c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_differentialevolution.py:487: UserWarning: differential_evolution: the 'workers' keyword has overridden updating='immediate' to updating='deferred'
  with DifferentialEvolutionSolver(func, bounds, args=args,

2024-11-06 15:46:26,565 [WARNING] Lost connection to localhost:9500.
2024-11-06 15:46:26,566 [WARNING] Lost connection to localhost:9503.
2024-11-06 15:46:26,567 [WARNING] Lost connection to localhost:9502.
2024-11-06 15:46:27,442 [INFO] Client 1949739997680-15268-624 connected to localhost:9500
2024-11-06 15:46:27,938 [INFO] Client 1949739996960-15268-624 connected to localhost:9501
2024-11-06 15:46:28,453 [INFO] Client 1949739996240-15268-624 connected to localhost:9503

c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0

Shutting down workers...

c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0
c:\Users\Latour\AppData\Local\miniforge-pypy3\envs\testim8\lib\site-packages\scipy\optimize\_numdiff.py:590: RuntimeWarning: invalid value encountered in subtract
  df = fun(x) - f0

({'X0': 0.10298154858669142, 'mu_max': 0.49542263458041547},
              message: Optimization terminated successfully.
              success: True
                  fun: 3.8587641855787576
                    x: [ 1.030e-01  4.954e-01]
                  nit: 24
                 nfev: 813
           population: [[ 1.030e-01  4.954e-01]
                        [ 1.015e-01  4.971e-01]
                        ...
                        [ 9.591e-02  5.050e-01]
                        [ 9.981e-02  4.991e-01]]
  population_energies: [ 3.859e+00  3.863e+00 ...  3.896e+00  3.875e+00])
Contents