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35f2be5 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 | """Correctness checks for the analytic kernels."""
import jax
import jax.numpy as jnp
import numpy as np
import pytest
jax.config.update("jax_enable_x64", True)
from ballast.kernels import (
HelmParams,
k_ext_full,
k_helm_mat,
k_thelm_mat,
matern32_blocks,
)
P = HelmParams(
phi_ls=0.8, phi_var=0.5, psi_ls=0.5, psi_var=0.5, time_ls=2.5, time_var=1.0
)
def _m32(t, t2, ls, var):
"""Reference Matern-3/2 written so autodiff is valid (no clipped distance)."""
r = jnp.abs(t - t2)
lam = jnp.sqrt(3.0) / ls
return var * (1 + lam * r) * jnp.exp(-lam * r)
def test_matern32_derivatives_match_autodiff():
"""Analytic d_t, d_t', d^2_tt' blocks agree with autodiff away from t=t'."""
ts = jnp.array([0.0, 0.3, 1.7, 4.2])
ts2 = jnp.array([0.5, 2.1, 3.3])
M = matern32_blocks(ts, ts2, P.time_ls, P.time_var)
k = jax.vmap(lambda a: jax.vmap(lambda b: _m32(a, b, P.time_ls, P.time_var))(ts2))(ts)
dt = jax.grad(_m32, argnums=0)
dt2 = jax.grad(_m32, argnums=1)
d2 = jax.grad(jax.grad(_m32, argnums=0), argnums=1)
f = lambda g: jax.vmap(lambda a: jax.vmap(lambda b: g(a, b, P.time_ls, P.time_var))(ts2))(ts)
np.testing.assert_allclose(M[..., 0, 0], k, rtol=1e-10)
np.testing.assert_allclose(M[..., 1, 0], f(dt), rtol=1e-8)
np.testing.assert_allclose(M[..., 0, 1], f(dt2), rtol=1e-8)
np.testing.assert_allclose(M[..., 1, 1], f(d2), rtol=1e-8)
def test_matern32_second_derivative_at_zero():
"""Paper Sec. H.2: d^2_{tt'} k at t=t' must be 3 sigma^2/l^2, not 0.
This is the value GPJax-style clipped-distance autodiff silently returns as
zero; it is also exactly P_inf[1,1] in the SPDE formulation, i.e. Var(d_t f).
"""
t = jnp.array([1.0])
M = matern32_blocks(t, t, 1.0, 1.0)
assert np.isclose(M[0, 0, 1, 1], 3.0)
M2 = matern32_blocks(t, t, P.time_ls, P.time_var)
assert np.isclose(M2[0, 0, 1, 1], 3.0 * P.time_var / P.time_ls**2)
def test_matern32_finite_difference():
"""Independent finite-difference check of the mixed second derivative."""
# h chosen to balance the O(h^2) truncation error of the 4-point stencil
# against O(eps/h^2) round-off; 1e-5 leaves ~1e-4 relative truncation error
# on a value of ~5e-3 and would fail a 1e-4 tolerance for benign reasons.
h = 1e-3
a, b = 1.0, 2.4
num = (
_m32(a + h, b + h, P.time_ls, P.time_var)
- _m32(a + h, b - h, P.time_ls, P.time_var)
- _m32(a - h, b + h, P.time_ls, P.time_var)
+ _m32(a - h, b - h, P.time_ls, P.time_var)
) / (4 * h * h)
ana = matern32_blocks(jnp.array([a]), jnp.array([b]), P.time_ls, P.time_var)[0, 0, 1, 1]
assert np.isclose(num, ana, rtol=1e-4)
def _helm_field_cov_reference(S, S2, p):
"""Helmholtz kernel rebuilt by autodiff of the potential/stream construction.
F_1 = d_1 Phi + d_2 Psi, F_2 = d_2 Phi - d_1 Psi, so
Cov(F_a(x), F_b(x')) is assembled from mixed second derivatives of k_Phi, k_Psi.
"""
kphi = lambda x, y: p.phi_var * jnp.exp(-0.5 * jnp.sum((x - y) ** 2) / p.phi_ls**2)
kpsi = lambda x, y: p.psi_var * jnp.exp(-0.5 * jnp.sum((x - y) ** 2) / p.psi_ls**2)
Hphi = jax.jacfwd(jax.jacrev(kphi, argnums=0), argnums=1)
Hpsi = jax.jacfwd(jax.jacrev(kpsi, argnums=0), argnums=1)
def cov(x, y):
A, B = Hphi(x, y), Hpsi(x, y)
return jnp.array(
[
[A[0, 0] + B[1, 1], A[0, 1] - B[1, 0]],
[A[1, 0] - B[0, 1], A[1, 1] + B[0, 0]],
]
)
return jax.vmap(lambda x: jax.vmap(lambda y: cov(x, y))(S2))(S)
def test_helmholtz_matches_autodiff_construction():
S = jnp.array([[0.0, 0.0], [0.4, -0.7], [1.2, 0.3]])
S2 = jnp.array([[0.1, 0.2], [-0.5, 0.9]])
ref = _helm_field_cov_reference(S, S2, P) # (3, 2, 2, 2)
got = k_helm_mat(S, S2, P).reshape(3, 2, 2, 2).transpose(0, 2, 1, 3)
# got is (i, j, c, c') after the transpose of the (i,c,j,c') flattening
np.testing.assert_allclose(got.transpose(0, 1, 2, 3), ref, rtol=1e-8, atol=1e-12)
def test_gram_matrices_are_psd():
key = jax.random.PRNGKey(0)
S = jax.random.uniform(key, (12, 2), minval=-2, maxval=2)
t = jnp.linspace(0, 5, 12)
for K in (k_thelm_mat(S, t, S, t, P), k_ext_full(S, t, P), k_helm_mat(S, S, P)):
w = jnp.linalg.eigvalsh(0.5 * (K + K.T))
assert w.min() > -1e-8, f"min eigenvalue {w.min()}"
def test_extended_kernel_blocks_are_consistent():
"""The f-block of the extended kernel is exactly the plain kernel."""
key = jax.random.PRNGKey(1)
S = jax.random.uniform(key, (5, 2), minval=-1, maxval=1)
t = jnp.linspace(0, 3, 5)
E = k_ext_full(S, t, P).reshape(5, 2, 2, 5, 2, 2)
plain = k_thelm_mat(S, t, S, t, P).reshape(5, 2, 5, 2)
np.testing.assert_allclose(E[:, :, 0, :, :, 0], plain, rtol=1e-10)
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