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kolmogorov_methods.py
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kolmogorov_methods.py
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# Copyright 2021 Google LLC
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# https://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
# ==============================================================================
import typing
from typing import Union, Tuple, Callable, NewType
import numpy as np
import jax
import jax.numpy as jnp
import flax.nn as nn
from dynamical_system import KolmogorovFlow
from util import aa_tuple_to_jnp, jnp_to_aa_tuple
Array = Union[np.ndarray, jnp.ndarray]
PrngKey = NewType('PrngKey', jnp.ndarray)
def generate_data_kolmogorov(
prng_key: PrngKey,
dyn_sys: KolmogorovFlow,
num_samples: int,
num_time_steps: int,
num_warmup_steps: int,
) -> Tuple[Array, Array, Array, Array]:
"""
Generates data for the Kolmogorov Flow model.
Args:
prng_key: key for random number generation.
dyn_sys: KolmogorovFlow dynamical system.
num_samples: number of independent samples to generate.
num_times_steps: number of snapshots to generate; the number of inner
integration steps is specified with the dynamical system instance.
num_warmup_steps: number of warmup steps.
Returns:
X0: initial state after warmup.
X: trajectory of physical states.
Y: trajectory of observed states.
offsets: offsets for AlignedArray data structure.
"""
X0_keys = jax.random.split(prng_key, num_samples)
X0 = dyn_sys.generate_filtered_velocity_fields(X0_keys)
total_warm_up_steps = num_warmup_steps * dyn_sys.num_inner_steps
X0 = dyn_sys.batch_warmup(X0, total_warm_up_steps)
X = dyn_sys.batch_integrate(X0, num_time_steps)
Y = dyn_sys.batch_observe(X)
return X0, X, Y, dyn_sys.offsets
def interpolate_periodic_kolmogorov(
u: Array,
factor: int,
method: str = 'bicubic',
) -> Array:
"""
Upsamples velocity field(s) `u` by `factor` under
the assumption that `u` is periodic in both upsampling dimensions.
Args:
u: jax.numpy.DeviceArray of shape (..., grid_x, grid_y, 2).
factor: scalar factor by which to resize grid_x and grid_y.
Returns:
Resized version of the velocity field(s).
"""
paddings = [(0,0)] * u.ndim
paddings[-2] = (1,1)
paddings[-3] = (1,1)
u_pad = jnp.pad(u, paddings, 'wrap')
out_shape = list(u_pad.shape)
out_shape[-2] = int(factor * out_shape[-2])
out_shape[-3] = int(factor * out_shape[-3])
out = jax.image.resize(u_pad, shape=out_shape, method=method)
fi = int(factor)
return out[..., fi:-fi, fi:-fi, :]
def interpolation_da_init_kolmogorov(
dyn_sys: KolmogorovFlow,
X0: Array,
) -> Array:
"""
Generates initial conditions for data assimilation by copying the observed
grid points and inferring the unobserved grid points as an average over
the dataset samples.
Args:
dyn_sys: DynamicalSystem.
X0: ground truth initial conditions of
shape (num_samples, grid_x, grid_y, 2).
Returns:
Initial conditions with observed grid points and otherwise sample
averaged grid points.
"""
Y0 = dyn_sys.batch_observe(X0)
factor = X0.shape[-2] / Y0.shape[-2]
X0_init = interpolate_periodic_kolmogorov(Y0, factor, method='bicubic')
return X0_init