Multi-Output & Time-Series

gsax accepts scalar, multi-output, and time-series multi-output arrays from the same API. This page uses one concrete model to show both (N, K) and (N, T, K) layouts.

Fully runnable example

import jax.numpy as jnp
import numpy as np
import gsax

problem = gsax.Problem.from_dict(
    {
        "amplitude": (0.5, 2.0),
        "frequency": (1.0, 5.0),
        "damping": (0.01, 0.5),
        "offset": (-1.0, 1.0),
    },
    output_names=("displacement", "velocity"),
)

time_values = np.linspace(0.0, 5.0, 40)


def oscillator_model(X):
    amp = X[:, 0, None]
    freq = X[:, 1, None]
    damping = X[:, 2, None]
    offset = X[:, 3, None]
    tt = jnp.asarray(time_values)[None, :]

    displacement = (
        amp * jnp.sin(2 * jnp.pi * freq * tt) * jnp.exp(-damping * tt) + offset
    )
    velocity = amp * jnp.cos(2 * jnp.pi * freq * tt) * jnp.exp(-damping * tt)

    return jnp.stack([displacement, velocity], axis=-1)  # (N, T, K=2)


sampling_result = gsax.sample(problem, n_samples=2048, seed=42)
X = jnp.asarray(sampling_result.samples)

Y_time = oscillator_model(X)      # (N, T, K)
Y_snapshot = Y_time[:, -1, :]     # (N, K)

time_result = gsax.analyze(sampling_result, Y_time)
snapshot_result = gsax.analyze(sampling_result, Y_snapshot)

print("Time-series S1 shape:", time_result.S1.shape)      # (T, K, D)
print("Time-series ST shape:", time_result.ST.shape)      # (T, K, D)
print("Snapshot S1 shape:", snapshot_result.S1.shape)     # (K, D)
print("Snapshot ST shape:", snapshot_result.ST.shape)     # (K, D)

print("Displacement sensitivities at the final time step:")
print(time_result.S1[-1, 0, :])

print("Velocity sensitivities for the snapshot:")
print(snapshot_result.S1[1, :])

Shape rules

• (N,) means scalar output.
• (N, K) means multiple outputs with no time dimension.
• (N, T, K) means time-series multi-output.
• A 2D array is always treated as (N, K), never (N, T).
• For a time-series with one output, reshape to (N, T, 1).

Single-output edge case

# Scalar output
Y_scalar = Y_snapshot[:, 0]      # (N,)
scalar_result = gsax.analyze(sampling_result, Y_scalar)
print(scalar_result.S1.shape)    # (D,)

# Time-series with one output
Y_one_output = Y_time[:, :, :1]  # (N, T, 1)
one_output_result = gsax.analyze(sampling_result, Y_one_output)
print(one_output_result.S1.shape)  # (T, 1, D)

Practical caveats

• Named outputs come from problem.output_names, so set them up early if you plan to export with to_dataset().
• calc_second_order=False removes S2, which can be a useful tradeoff for large (T, K) outputs when you only need S1 and ST.
• The same shape rules apply to gsax.analyze_hdmr().

See also

• xarray Labeled Output for named access by parameter, output, and time coordinate.
• RS-HDMR Example for the same shape rules on the surrogate workflow.
• Advanced Workflow for a bigger custom model that combines Sobol, HDMR, emulator prediction, and dataset export.