def simulate_rom(trainsize, r, regs, steps=None):
"""Load everything needed to simulate a given ROM, run the simulation,
and return the simulation results and everything needed to reconstruct
the results in the original high-dimensional space.
Raise an Exception if any of the ingredients are missing.
Parameters
----------
trainsize : int
Number of snapshots used to train the ROM.
r : int
Dimension of the ROM.
regs : two or three positive floats
Regularization hyperparameters used to train the ROM.
steps : int or None
Number of time steps to simulate the ROM.
Returns
-------
t : (nt,) ndarray
Time domain corresponding to the ROM outputs.
V : (NUM_ROMVARS*DOF,r) ndarray
POD basis used to project the training data (and for reconstructing
the full-order scaled predictions).
qbar : (NUM_ROMVARS*DOF,) ndarray
Mean snapshot that the training data was shifted by after scaling
but before projection.
scales : (NUM_ROMVARS,4) ndarray
Information for how the data was scaled. See data_processing.scale().
q_rom : (nt,r) ndarray
Prediction results from the ROM.
"""
# Load the time domain, basis, initial conditions, and trained ROM.
t = utils.load_time_domain(steps)
V, qbar, scales = utils.load_basis(trainsize, r)
Q_, _, _ = utils.load_projected_data(trainsize, r)
rom = utils.load_rom(trainsize, r, regs)
# Simulate the ROM over the full time domain.
with utils.timed_block(f"Simulating ROM with k={trainsize:d}, r={r:d}, "
f"{config.REGSTR(regs)} over full time domain"):
q_rom = rom.predict(Q_[:, 0], t, config.U, method="RK45")
return t, V, qbar, scales, q_rom
def simulate_rom(trainsize, r, reg, steps=None):
"""Load everything needed to simulate a given ROM, simulate the ROM,
and return the simulation results and everything needed to reconstruct
the results in the original high-dimensional space.
Raise an Exception if any of the ingredients are missing.
Parameters
----------
trainsize : int
Number of snapshots used to train the ROM.
r : int
Dimension of the ROM. This is also the number of retained POD
modes (left singular vectors) used to project the training data.
reg : float
Regularization parameter used to train the ROM.
steps : int or None
Number of time steps to simulate the ROM.
Returns
-------
t : (nt,) ndarray
Time domain corresponding to the ROM outputs.
V : (config*NUM_ROMVARS*config.DOF,r) ndarray
POD basis used to project the training data (and for reconstructing
the full-order scaled predictions).
scales : (NUM_ROMVARS,4) ndarray
Information for how the data was scaled. See data_processing.scale().
x_rom : (nt,r) ndarray
Prediction results from the ROM.
"""
# Load the time domain, basis, initial conditions, and trained ROM.
t = utils.load_time_domain(steps)
V, _ = utils.load_basis(trainsize, r)
X_, _, _, scales = utils.load_projected_data(trainsize, r)
rom = utils.load_rom(trainsize, r, reg)
# Simulate the ROM over the full time domain.
with utils.timed_block(f"Simulating ROM with r={r:d}, "
f"reg={reg:e} over full time domain"):
x_rom = rom.predict(X_[:, 0], t, config.U, method="RK45")
return t, V, scales, x_rom
def main(timeindices,
variables=None,
snaptype=["gems", "rom", "error"],
trainsize=None,
r=None,
reg=None):
"""Convert a snapshot in .h5 format to a .dat file that matches the format
of grid.dat. The new file is saved in `config.tecplot_path()` with the same
filename and the new file extension .dat.
Parameters
----------
timeindices : ndarray(int) or int
Indices (one-based) in the full time domain of the snapshots to save.
variables : str or list(str)
The variables to scale, a subset of config.ROM_VARIABLES.
Defaults to all variables.
snaptype : {"rom", "gems", "error"} or list(str)
Which kinds of snapshots to save. Options:
* "gems": snapshots from the full-order GEMS data;
* "rom": reconstructed snapshots produced by a ROM;
* "error": absolute error between the full-order data
and the reduced-order reconstruction.
If "rom" or "error" are selected, the ROM is selected by the
remaining arguments.
trainsize : int
Number of snapshots used to train the ROM.
r : int
Number of retained modes in the ROM.
reg : float
Regularization factor used to train the ROM.
"""
utils.reset_logger(trainsize)
# Parse parameters.
timeindices = np.sort(np.atleast_1d(timeindices))
simtime = timeindices.max()
t = utils.load_time_domain(simtime + 1)
# Parse the variables.
if variables is None:
variables = config.ROM_VARIABLES
elif isinstance(variables, str):
variables = [variables]
varnames = '\n'.join(f'"{v}"' for v in variables)
if isinstance(snaptype, str):
snaptype = [snaptype]
for stype in snaptype:
if stype not in ("gems", "rom", "error"):
raise ValueError(f"invalid snaptype '{stype}'")
# Read the grid file.
with utils.timed_block("Reading Tecplot grid data"):
# Parse the header.
grid_path = config.grid_data_path()
with open(grid_path, 'r') as infile:
grid = infile.read()
if int(re.findall(r"Elements=(\d+)", grid)[0]) != config.DOF:
raise RuntimeError(f"{grid_path} DOF and config.DOF do not match")
num_nodes = int(re.findall(r"Nodes=(\d+)", grid)[0])
end_of_header = re.findall(r"DT=.*?\n", grid)[0]
headersize = grid.find(end_of_header) + len(end_of_header)
# Extract geometry information.
grid_data = grid[headersize:].split()
x = grid_data[:num_nodes]
y = grid_data[num_nodes:2 * num_nodes]
cell_volume = grid_data[2 * num_nodes:3 * num_nodes]
connectivity = grid_data[3 * num_nodes:]
# Extract full-order data if needed.
if ("gems" in snaptype) or ("error" in snaptype):
gems_data, _ = utils.load_gems_data(cols=timeindices)
with utils.timed_block("Lifting selected snapshots of GEMS data"):
lifted_data = dproc.lift(gems_data)
true_snaps = np.concatenate(
[dproc.getvar(v, lifted_data) for v in variables])
# Simulate ROM if needed.
if ("rom" in snaptype) or ("error" in snaptype):
# Load the SVD data.
V, _ = utils.load_basis(trainsize, r)
# Load the initial conditions and scales.
X_, _, _, scales = utils.load_projected_data(trainsize, r)
# Load the appropriate ROM.
rom = utils.load_rom(trainsize, r, reg)
# Simulate the ROM over the time domain.
with utils.timed_block(f"Simulating ROM with r={r:d}, reg={reg:.0e}"):
x_rom = rom.predict(X_[:, 0], t, config.U, method="RK45")
if np.any(np.isnan(x_rom)) or x_rom.shape[1] < simtime:
raise ValueError("ROM unstable!")
# Reconstruct the results (only selected variables / snapshots).
with utils.timed_block("Reconstructing simulation results"):
x_rec = dproc.unscale(V[:, :r] @ x_rom[:, timeindices], scales)
x_rec = np.concatenate([dproc.getvar(v, x_rec) for v in variables])
dsets = {}
if "rom" in snaptype:
dsets["rom"] = x_rec
if "gems" in snaptype:
dsets["gems"] = true_snaps
if "error" in snaptype:
with utils.timed_block("Computing absolute error of reconstruction"):
abs_err = np.abs(true_snaps - x_rec)
dsets["error"] = abs_err
# Save each of the selected snapshots in Tecplot format matching grid.dat.
for j, tindex in enumerate(timeindices):
header = HEADER.format(varnames, tindex, t[tindex], num_nodes,
config.DOF,
len(variables) + 2, "SINGLE " * len(variables))
for label, dset in dsets.items():
if label == "gems":
save_path = config.gems_snapshot_path(tindex)
if label in ("rom", "error"):
folder = config.rom_snapshot_path(trainsize, r, reg)
save_path = os.path.join(folder, f"{label}_{tindex:05d}.dat")
with utils.timed_block(f"Writing {label} snapshot {tindex:05d}"):
with open(save_path, 'w') as outfile:
# Write the header.
outfile.write(header)
# Write the geometry data (x,y coordinates).
for i in range(0, len(x), NCOLS):
outfile.write(' '.join(x[i:i + NCOLS]) + '\n')
for i in range(0, len(y), NCOLS):
outfile.write(' '.join(y[i:i + NCOLS]) + '\n')
# Write the data for each variable.
for i in range(0, dset.shape[0], NCOLS):
row = ' '.join(f"{v:.9E}"
for v in dset[i:i + NCOLS, j])
outfile.write(row + '\n')
# Write connectivity information.
for i in range(0, len(connectivity), NCOLS):
outfile.write(' '.join(connectivity[i:i + NCOLS]) +
'\n')
def _train_minimize_1D(trainsize, r, regs, testsize=None, margin=1.1):
"""Train ROMs with the given dimension(s), saving only the ROM with
the least training error that satisfies a bound on the integrated POD
coefficients, using a search algorithm to choose the regularization
parameter.
Parameters
----------
trainsize : int
Number of snapshots to use to train the ROM.
r : int
Dimension of the desired ROM. Also the number of retained POD modes
(left singular vectors) used to project the training data.
regs : two non-negative floats
Bounds for the (single) regularization hyperparameter to use in the
Operator Inference least-squares problem for training the ROM.
testsize : int
Number of time steps for which a valid ROM must satisfy the POD bound.
margin : float ≥ 1
Amount that the integrated POD coefficients of a valid ROM are allowed
to deviate in magnitude from the maximum magnitude of the training
data Q, i.e., bound = margin * max(abs(Q)).
"""
utils.reset_logger(trainsize)
# Parse aguments.
check_lstsq_size(trainsize, r, modelform="cAHB")
log10regs = np.log10(regs)
# Load training data.
t = utils.load_time_domain(testsize)
Q_, Qdot_, _ = utils.load_projected_data(trainsize, r)
U = config.U(t[:trainsize])
# Compute the bound to require for integrated POD modes.
B = margin * np.abs(Q_).max()
# Create a solver mapping regularization hyperparameters to operators.
with utils.timed_block(f"Constructing least-squares solver, r={r:d}"):
rom = opinf.InferredContinuousROM("cAHB")
rom._construct_solver(None, Q_, Qdot_, U, 1)
# Test each regularization hyperparameter.
def training_error(log10reg):
"""Return the training error resulting from the regularization
hyperparameters λ1 = λ2 = 10^log10reg. If the resulting model
violates the POD bound, return "infinity".
"""
λ = 10**log10reg
# Train the ROM on all training snapshots.
with utils.timed_block(f"Testing ROM with λ={λ:e}"):
rom._evaluate_solver(λ)
# Simulate the ROM over the full domain.
with np.warnings.catch_warnings():
np.warnings.simplefilter("ignore")
q_rom = rom.predict(Q_[:, 0], t, config.U, method="RK45")
# Check for boundedness of solution.
if not is_bounded(q_rom, B):
return _MAXFUN
# Calculate integrated relative errors in the reduced space.
return opinf.post.Lp_error(Q_, q_rom[:, :trainsize],
t[:trainsize])[1]
opt_result = opt.minimize_scalar(training_error,
method="bounded",
bounds=log10regs)
if opt_result.success and opt_result.fun != _MAXFUN:
λ = 10**opt_result.x
with utils.timed_block(f"Best regularization for k={trainsize:d}, "
f"r={r:d}: λ={λ:.0f}"):
rom._evaluate_solver(λ)
save_trained_rom(trainsize, r, (λ, λ), rom)
else:
message = "Regularization search optimization FAILED"
print(message)
logging.info(message)
def train_gridsearch(trainsize, r, regs, testsize=None, margin=1.1):
"""Train ROMs with the given dimension over a grid of potential
regularization hyperparameters, saving only the ROM with the least
training error that satisfies a bound on the integrated POD coefficients.
Parameters
----------
trainsize : int
Number of snapshots to use to train the ROM.
r : int
Dimension of the desired ROM. Also the number of retained POD modes
(left singular vectors) used to project the training data.
regs : (float, float, int, float, float, int)
Bounds and sizes for the grid of regularization hyperparameters.
First-order: search in [regs[0], regs[1]] at regs[2] points.
Quadratic: search in [regs[3], regs[4]] at regs[5] points.
Cubic: search in [regs[6], regs[7]] at regs[8] points.
testsize : int
Number of time steps for which a valid ROM must satisfy the POD bound.
margin : float ≥ 1
Amount that the integrated POD coefficients of a valid ROM are allowed
to deviate in magnitude from the maximum magnitude of the training
data Q, i.e., bound = margin * max(abs(Q)).
"""
utils.reset_logger(trainsize)
# Parse aguments.
if len(regs) not in [6, 9]:
raise ValueError("6 or 9 regs required (bounds / sizes of grids")
grids = []
for i in range(0, len(regs), 3):
check_regs(regs[i:i + 2])
grids.append(
np.logspace(np.log10(regs[i]), np.log10(regs[i + 1]),
int(regs[i + 2])))
modelform = get_modelform(grids)
d = check_lstsq_size(trainsize, r, modelform)
# Load training data.
t = utils.load_time_domain(testsize)
Q_, Qdot_, _ = utils.load_projected_data(trainsize, r)
U = config.U(t[:trainsize])
# Compute the bound to require for integrated POD modes.
M = margin * np.abs(Q_).max()
# Create a solver mapping regularization hyperparameters to operators.
num_tests = np.prod([grid.size for grid in grids])
print(f"TRAINING {num_tests} ROMS")
with utils.timed_block(f"Constructing least-squares solver, r={r:d}"):
rom = opinf.InferredContinuousROM(modelform)
rom._construct_solver(None, Q_, Qdot_, U, np.ones(d))
# Test each regularization hyperparameter.
errors_pass = {}
errors_fail = {}
for i, regs in enumerate(itertools.product(*grids)):
with utils.timed_block(f"({i+1:d}/{num_tests:d}) Testing ROM with "
f"{config.REGSTR(regs)}"):
# Train the ROM on all training snapshots.
rom._evaluate_solver(regularizer(r, *list(regs)))
# Simulate the ROM over the full domain.
with np.warnings.catch_warnings():
np.warnings.simplefilter("ignore")
q_rom = rom.predict(Q_[:, 0], t, config.U, method="RK45")
# Check for boundedness of solution.
errors = errors_pass if is_bounded(q_rom, M) else errors_fail
# Calculate integrated relative errors in the reduced space.
if q_rom.shape[1] > trainsize:
errors[tuple(regs)] = opinf.post.Lp_error(
Q_, q_rom[:, :trainsize], t[:trainsize])[1]
# Choose and save the ROM with the least error.
if not errors_pass:
message = f"NO STABLE ROMS for r={r:d}"
print(message)
logging.info(message)
return
err2reg = {err: reg for reg, err in errors_pass.items()}
regs = list(err2reg[min(err2reg.keys())])
with utils.timed_block(f"Best regularization for k={trainsize:d}, "
f"r={r:d}: {config.REGSTR(regs)}"):
rom._evaluate_solver(regularizer(r, *regs))
save_trained_rom(trainsize, r, regs, rom)
def train_with_minimization(trainsize, num_modes, regs,
testsize=None, margin=1.5):
"""Train ROMs with the given dimension(s), saving only the ROM with
the least training error that satisfies a bound on the integrated POD
coefficients, using a search algorithm to choose the regularization
parameter.
Parameters
----------
trainsize : int
Number of snapshots to use to train the ROM(s).
num_modes : int or list(int)
Dimension of the ROM(s) to train, i.e., the number of retained POD
modes (left singular vectors) used to project the training data.
regs : [float, float]
regularization parameter(s) to use in the training.
testsize : int
Number of time steps for which a valid ROM must satisfy the POD bound.
margin : float >= 1
Amount that the integrated POD coefficients of a valid ROM are allowed
to deviate in magnitude from the maximum magnitude of the training
data Q, i.e., bound = margin * max(abs(Q)).
"""
utils.reset_logger(trainsize)
# Parse aguments.
if np.isscalar(num_modes):
num_modes = [num_modes]
if np.isscalar(regs) or len(regs) != 2:
raise ValueError("2 regularizations required (reg_low, reg_high)")
bounds = np.log10(regs)
# Load the full time domain and evaluate the input function.
t = utils.load_time_domain(testsize)
Us = config.U(t)
for r in num_modes:
# Load training data.
X_, Xdot_, _, scales = utils.load_projected_data(trainsize, r)
# Compute the bound to require for integrated POD modes.
B = margin * np.abs(X_).max()
# Test each regularization parameter.
def training_error_from_rom(log10reg):
reg = 10**log10reg
# Train the ROM on all training snapshots.
with utils.timed_block(f"Testing ROM with r={r:d}, reg={reg:e}"):
rom = train_rom(X_, Xdot_, Us[:trainsize], reg)
if not rom:
return _MAXFUN
# Simulate the ROM over the full domain.
with np.warnings.catch_warnings():
np.warnings.simplefilter("ignore")
x_rom = rom.predict(X_[:,0], t, config.U, method="RK45")
# Check for boundedness of solution.
if not is_bounded(x_rom, B):
return _MAXFUN
# Calculate integrated relative errors in the reduced space.
return roi.post.Lp_error(X_, x_rom[:,:trainsize],
t[:trainsize])[1]
opt_result = opt.minimize_scalar(training_error_from_rom,
bounds=bounds, method="bounded")
if opt_result.success and opt_result.fun != _MAXFUN:
best_reg = 10 ** opt_result.x
best_rom = train_rom(X_, Xdot_, Us[:trainsize], best_reg)
save_best_trained_rom(trainsize, r, best_reg, best_rom)
else:
print(f"Regularization search optimization FAILED for r = {r:d}")
def train_with_gridsearch(trainsize, num_modes, regs,
testsize=None, margin=1.5):
"""Train ROMs with the given dimension(s) and regularization(s),
saving only the ROM with the least training error that satisfies
a bound on the integrated POD coefficients.
Parameters
----------
trainsize : int
Number of snapshots to use to train the ROM(s).
num_modes : int or list(int)
Dimension of the ROM(s) to train, i.e., the number of retained POD
modes (left singular vectors) used to project the training data.
regs : float or list(float)
regularization parameter(s) to use in the training.
testsize : int
Number of time steps for which a valid ROM must satisfy the POD bound.
margin : float >= 1
Amount that the integrated POD coefficients of a valid ROM are allowed
to deviate in magnitude from the maximum magnitude of the training
data Q, i.e., bound = margin * max(abs(Q)).
"""
utils.reset_logger(trainsize)
# Parse aguments.
if np.isscalar(num_modes):
num_modes = [num_modes]
if np.isscalar(regs):
regs = [regs]
# Load the full time domain and evaluate the input function.
t = utils.load_time_domain(testsize)
Us = config.U(t)
logging.info(f"TRAINING {len(num_modes)*len(regs)} ROMS")
for ii,r in enumerate(num_modes):
# Load training data.
X_, Xdot_, _, scales = utils.load_projected_data(trainsize, r)
# Compute the bound to require for integrated POD modes.
M = margin * np.abs(X_).max()
# Test each regularization parameter.
trained_roms = {}
errors_pass = {}
errors_fail = {}
for reg in regs:
# Train the ROM on all training snapshots.
with utils.timed_block(f"Testing ROM with r={r:d}, reg={reg:e}"):
rom = train_rom(X_, Xdot_, Us[:trainsize], reg)
if not rom:
continue # Skip if training fails.
trained_roms[reg] = rom
# Simulate the ROM over the full domain.
with np.warnings.catch_warnings():
np.warnings.simplefilter("ignore")
x_rom = rom.predict(X_[:,0], t, config.U, method="RK45")
# Check for boundedness of solution.
errors = errors_pass if is_bounded(x_rom, M) else errors_fail
# Calculate integrated relative errors in the reduced space.
if x_rom.shape[1] > trainsize:
errors[reg] = roi.post.Lp_error(X_, x_rom[:,:trainsize],
t[:trainsize])[1]
# Choose and save the ROM with the least error.
plt.semilogx(list(errors_fail.keys()), list(errors_fail.values()),
f"C{ii}x", mew=1, label=fr"$r = {r:d}$, bound violated")
if not errors_pass:
print(f"NO STABLE ROMS for r = {r:d}")
continue
err2reg = {err:reg for reg,err in errors_pass.items()}
best_reg = err2reg[min(err2reg.keys())]
best_rom = trained_roms[best_reg]
save_best_trained_rom(trainsize, r, best_reg, best_rom)
plt.semilogx(list(errors_pass.keys()), list(errors_pass.values()),
f"C{ii}*", mew=0, label=fr"$r = {r:d}$, bound satisfied")
plt.axvline(best_reg, lw=.5, color=f"C{ii}")
plt.legend()
plt.xlabel(r"Regularization parameter $\lambda$")
plt.ylabel(r"ROM relative error $\frac"
r"{||\widehat{\mathbf{Q}} - \widetilde{\mathbf{Q}}'||}"
r"{||\widehat{\mathbf{Q}}||}$")
plt.ylim(0, 1)
plt.xlim(min(regs), max(regs))
plt.title(fr"$n_t = {trainsize}$")
utils.save_figure(f"regsweep_nt{trainsize:05d}.pdf")
def train_gridsearch(trainsize, r, regs, testsize=None, margin=1.5):
"""Train ROMs with the given dimension over a grid of potential
regularization hyperparameters, saving only the ROM with the least
training error that satisfies a bound on the integrated POD coefficients.
Parameters
----------
trainsize : int
Number of snapshots to use to train the ROM.
r : int
Dimension of the desired ROM. Also the number of retained POD modes
(left singular vectors) used to project the training data.
regs : (float, float, int, float, float, int)
Bounds and sizes for the grid of regularization parameters.
Linear: search in [regs[0], regs[1]] at regs[2] points.
Quadratic: search in [regs[3], regs[4]] at regs[5] points.
testsize : int
Number of time steps for which a valid ROM must satisfy the POD bound.
margin : float >= 1
Amount that the integrated POD coefficients of a valid ROM are allowed
to deviate in magnitude from the maximum magnitude of the training
data Q, i.e., bound = margin * max(abs(Q)).
"""
utils.reset_logger(trainsize)
# Parse aguments.
d = check_lstsq_size(trainsize, r)
if len(regs) != 6:
raise ValueError("len(regs) != 6 (bounds / sizes for parameter grid")
check_regs(regs[0:2])
check_regs(regs[3:5])
λ1grid = np.logspace(np.log10(regs[0]), np.log10(regs[1]), int(regs[2]))
λ2grid = np.logspace(np.log10(regs[3]), np.log10(regs[4]), int(regs[5]))
# Load training data.
t = utils.load_time_domain(testsize)
Q_, Qdot_, _ = utils.load_projected_data(trainsize, r)
U = config.U(t[:trainsize])
# Compute the bound to require for integrated POD modes.
M = margin * np.abs(Q_).max()
# Create a solver mapping regularization parameters to operators.
print(f"TRAINING {λ1grid.size*λ2grid.size} ROMS")
with utils.timed_block(f"Constructing least-squares solver, r={r:d}"):
rom = roi.InferredContinuousROM(config.MODELFORM)
rom._construct_solver(None, Q_, Qdot_, U, np.ones(d))
# Test each regularization parameter.
errors_pass = {}
errors_fail = {}
for λ1, λ2 in itertools.product(λ1grid, λ2grid):
with utils.timed_block(f"Testing ROM with λ1={λ1:5e}, λ2={λ2:5e}"):
# Train the ROM on all training snapshots.
rom._evaluate_solver(regularizer(r, d, λ1, λ2))
# Simulate the ROM over the full domain.
with np.warnings.catch_warnings():
np.warnings.simplefilter("ignore")
q_rom = rom.predict(Q_[:, 0], t, config.U, method="RK45")
# Check for boundedness of solution.
errors = errors_pass if is_bounded(q_rom, M) else errors_fail
# Calculate integrated relative errors in the reduced space.
if q_rom.shape[1] > trainsize:
errors[(λ1, λ2)] = roi.post.Lp_error(Q_, q_rom[:, :trainsize],
t[:trainsize])[1]
# Choose and save the ROM with the least error.
if not errors_pass:
message = f"NO STABLE ROMS for r={r:d}"
print(message)
logging.info(message)
return
err2reg = {err: reg for reg, err in errors_pass.items()}
λ1, λ2 = err2reg[min(err2reg.keys())]
with utils.timed_block(f"Best regularization for k={trainsize:d}, "
f"r={r:d}: λ1={λ1:.0f}, λ2={λ2:.0f}"):
rom._evaluate_solver(regularizer(r, d, λ1, λ2))
save_trained_rom(trainsize, r, (λ1, λ2), rom)
