⚠️ Beta Release: FIPS is currently in beta (v0.1.0b1). The core API is stable and ready for testing, but may evolve based on user feedback. Please report any issues or edge cases you encounter.

Bridging Physics and Data

Inverse problems in geophysics and atmospheric science are incredibly complex, often involving massive state spaces, deeply heterogeneous observational networks, and explicit matrix-algebra requirements. While many general-purpose optimization tools exist, they often force researchers to strip away critical spatiotemporal metadata or translate pre-computed physical models into rigid, abstract array structures.

FIPS (Flexible Inverse Problem Solver) is built from the ground up to solve linear, matrix-based inverse problems without losing the context of your data. It provides the structural flexibility to handle messy, real-world realities seamlessly:

• Native N-Dimensional Alignment: FIPS natively utilizes pandas.MultiIndex to smoothly align heterogeneous datasets across any dimension. Whether you are mixing temporal, spatial, spectral, or sensor-specific data, your coordinates are never dropped or misaligned.

• Modular Block Architecture: Avoid wrangling monolithic arrays. Construct massive, multi-source state spaces and observation networks piece-by-piece using specialized Block and MatrixBlock objects.

• Speak Your Domain's Language: Built explicitly around the standard y = Hx + error paradigm. Directly plug in your pre-computed forward operators ($H$), prior covariances ($S_0$), and model-data mismatches ($S_z$).

• Analytical Speed & Sparse Support: FIPS is built for scale. By leveraging optimized sparse data structures and direct linear algebra rather than expensive sampling algorithms, it computes exact analytical Maximum A Posteriori (MAP) estimates for massive state spaces in seconds.

Installation

From GitHub

pip install git+https://github.com/jmineau/fips

From Source

git clone https://github.com/jmineau/fips.git
cd fips
pip install -e .

Usage

Single-block — one observation source

import numpy as np
import pandas as pd
from fips import Block, CovarianceMatrix
from fips.problems.flux import FluxProblem

# State: gridded prior fluxes (time × lat × lon)
flux_idx = pd.MultiIndex.from_product(
    [pd.date_range("2023-01", periods=3, freq="MS"), [37.0, 38.0], [-112.0, -111.0]],
    names=["time", "lat", "lon"],
)
prior = pd.Series(np.ones(12) * 1.5, index=flux_idx, name="flux")

# Observations: tower concentration measurements
obs_idx = pd.MultiIndex.from_product(
    [pd.date_range("2023-01", periods=8, freq="2W"), ["UOU"]],
    names=["time", "site"],
)
obs = pd.Series(np.ones(8) * 400.0, index=obs_idx, name="concentration")

# Forward operator (Jacobian), flux error covariance, obs error covariance
H   = pd.DataFrame(np.random.rand(8, 12), index=obs_idx,  columns=flux_idx)
S_0 = pd.DataFrame(np.eye(12) * 0.5,  index=flux_idx, columns=flux_idx)
S_z = pd.DataFrame(np.eye(8)  * 0.1,  index=obs_idx,  columns=obs_idx)

problem = FluxProblem(
    obs=obs, prior=prior,
    forward_operator=H, prior_error=S_0, modeldata_mismatch=S_z,
).solve()

print(problem.posterior_fluxes)        # posterior pd.Series indexed by (time, lat, lon)
print(problem.estimator.reduced_chi2)  # reduced chi-squared statistic

Multi-block — combined station + satellite observations

import numpy as np
import pandas as pd
from fips import Block, Vector, Matrix, MatrixBlock, CovarianceMatrix, InverseProblem

# State: same gridded prior fluxes (from above)
N_f = 12

# Obs block 1: ground station in-situ concentrations
station_idx = pd.MultiIndex.from_product(
    [pd.date_range("2023-01", periods=8, freq="2W"), ["UOU"]],
    names=["time", "site"],
)
station_obs = Block(pd.Series(np.ones(8) * 400.0,
                              index=station_idx, name="station"))

# Obs block 2: satellite column-average concentrations
satellite_idx = pd.MultiIndex.from_product(
    [pd.date_range("2023-01", periods=3, freq="MS"), [37.5], [-111.5]],
    names=["time", "lat", "lon"],
)
satellite_obs = Block(pd.Series(np.ones(3) * 0.00400,
                                index=satellite_idx, name="satellite"))

# Combine obs blocks into a list
obs_blks = [station_obs, satellite_obs]

# Jacobian: one MatrixBlock per obs type, both mapping to the "flux" state block
H_blks = [
    MatrixBlock(
        pd.DataFrame(np.random.rand(8, N_f),
        index=station_idx, columns=flux_idx),
        row_block="station", col_block="flux",
    ),
    MatrixBlock(
        pd.DataFrame(np.random.rand(3, N_f),
        index=satellite_idx, columns=flux_idx),
        row_block="satellite", col_block="flux",
    ),]

# Prior error covariance: only flux errors, no cross-block covariances
S_0 = CovarianceMatrix(np.eye(N_f) * 0.5, index=flux_idx, columns=flux_idx)

# Model-data mismatch covariance: block-diagonal with separate error levels for stations vs. satellite
S_z_blks = [
    CovarianceMatrix(np.eye(8) * 0.1, index=station_idx, columns=station_idx),
    CovarianceMatrix(np.eye(3) * 0.2, index=satellite_idx, columns=satellite_idx),
]

# Pass blocks to the InverseProblem and solve
problem = InverseProblem(
    obs=obs_blks, prior=prior,
    forward_operator=H_blks, prior_error=S_0, modeldata_mismatch=S_z_blks,
).solve()

print(problem.posterior['flux'])       # posterior pd.Series indexed by (time, lat, lon)
print(problem.estimator.reduced_chi2)  # reduced chi-squared statistic

Documentation

Full documentation is available at https://jmineau.github.io/fips/

Contributing

Contributions are welcome! Please see CONTRIBUTING.md for guidelines.