Model. Estimate. Quantify. Validate.

Otwin is a Python library used to build Digital Twins of physical equipment from physics, data or both, keep them in step with the machine from live measurements, and forecast with an uncertainty band whose coverage has been measured and a skill score earned out of sample against a reference that is hard to beat.

What otwin is for

A digital twin is a model of one particular physical asset (machine, pump, battery bank, heat exchanger, chemical reactor, a complete process) kept up to date from that asset's own measurements and run forward to support decissions about it.

Electrical grid. This distribution transformer keeps running above nameplate on hot afternoons. How much longer can it do that before the winding reaches its thermal limit?

Renewable generation and storage. This battery bank is three years into its life. How much can I commit to the market next week and still be certain of delivering it?

Water treatment. The transfer pumps need more power every month for the same flow. How many weeks of margin are left before the duty pump can no longer hold its setpoint?

How much capacity does this Li-ion battery still have? When will it cross the retirement threshold?

Choosing the model structure

Physical assets like real equipment or complete processess are rarely fully known or fully unknown. Otwin provides five model classes and the manifest records which one you used, because the guarantees available to you depend on the answer.

Digital Twins can be classified as:

	White box	Every equation and every parameter comes from first principles. Nothing is fitted. The guarantee is structural — and so is the limit: it can only describe what you can write down.
	Grey box	The structure is fixed by physics; the unknown parts are estimated from data. Almost every useful industrial twin is here.
	Black box	The data decides everything. Excellent inside the range it has seen, and no reason to behave outside it.

What makes otwin different

• The uncertainty is measured, not declared. A stated 90 % interval means nothing until somebody counts how often it contains the truth. Interval.is_validated stays False until it has been counted. The conformal example below finishes at a measured 90 % on sixty held-out cycles against a 90 % target; when the calibration set is too small to support the level requested, the library returns an infinite half-width, loudly, rather than a comfortable-looking one.

• The validation is leakage-free and beats something. Partitions are out-of-sample by default, a reference forecaster is mandatory, and the report leads with the skill score against the hardest of persistence, drift, mean and seasonal-naive. The model is handed history and a horizon, never the test values.

• The twin can say no. Operating range, forecast horizon and calibration status are recorded in the manifest, and a request outside them is refused with a reason rather than answered with a number.

Quick install

pip install otwin

Package-level tree

otwin/
├── io          read equipment or a dataset: SunSpec, Modbus, units, quality      [field]
├── signal      condition the series: resample, find gaps, report coverage
├── model       the physics: port-Hamiltonian, irreversible, catalogue, learned      [nn]
├── estimate    correct the state: EKF, bounded MHE, energy-consistent observer
├── forecast    predict and prove it: baselines, skill, conformal, ensembles         [gp]
├── advise      the validated envelope: answer, or refuse with a reason
└── interfaces  the contract the other six compose through — protocols, no algorithms

otwin/interfaces sits outside the chain: the six packages compose through protocols rather than by importing each other, so you can take one on its own — score a model that has nothing to do with otwin, or write physics and never touch io.

Get started

A white-box example consisting of a mass on a spring with a damper. The system has two energy stores: the spring and the moving mass.

import numpy as np
from otwin.model import PortHamiltonianSystem, integrate_phs

# State: x[0] = spring displacement q [m], x[1] = momentum p = m*v [kg m/s]
# k = 2 N/m, m = 1 kg, c = 0.3 N s/m
osc = PortHamiltonianSystem(
    H      = lambda x: 0.5 * 2.0 * x[0]**2 + 0.5 * x[1]**2,   # stored energy [J]
    grad_H = lambda x: np.array([2.0 * x[0], x[1]]),          # [force, velocity]
    J      = lambda x: np.array([[0.0, 1.0], [-1.0, 0.0]]),   # spring <-> mass exchange
    R      = lambda x: np.array([[0.0, 0.0], [0.0, 0.3]]),    # the damper
    g      = lambda x: np.array([[0.0], [1.0]]),              # external force on the mass
    n_states=2, n_inputs=1,
)

t   = np.linspace(0, 20, 400)      # 20 s
u   = np.zeros((400, 1))           # ports closed: no external force
sol = integrate_phs(osc, np.array([1.0, 0.0]), t, u)

E = np.array([osc.energy(x) for x in sol["x"]])
print(f"Stored energy: {E[0]:.4f} J at t=0  ->  {E[-1]:.4f} J at t=20 s")
print(f"Largest single-step energy INCREASE: {max(np.max(np.diff(E)), 0.0):.2e} J")

Stored energy: 1.0000 J at t=0  ->  0.0024 J at t=20 s
Largest single-step energy INCREASE: 0.00e+00 J

That second line Largest single-step energy INCREASE: 0.00e+00 J show that with no force applied, stored energy never rises because the model form makes it impossible.

When the physics is known (white-box)

Choose a port-Hamiltonian structure and conservation stops being something you check and becomes something you cannot violate: it is an algebraic property of how the model is written, so it survives any parameter set, any step size and any length of run.

In the first example below, stored energy does not rise on a single step of a 400-step simulation with the ports closed, where scipy.integrate.solve_ivp on the same right-hand side gains 3.2e-05 J. On the reference cases the same structure reproduces closed-form answers to the limit of double precision — a pumped-hydro round-trip efficiency agrees with its analytic value to a relative error of 9.6e-16 — and the irreversible form holds the second law across a full reactor run in which the same process written as a plain energy balance violates it on 91 % of steps.

What none of this gives you: accuracy. A model with the wrong parameters is still wrong — it is simply wrong without violating the energy balance. Accuracy is measured separately, and that is what the rest of the library is for.

Writing a first-principles model using Port Hamniltonian Systems

You describe the system with four functions of the state. The dynamics follow from them:

ẋ = dx/dt = ( J(x) - R(x) ) @ grad_H(x)  +  g(x) @ u        # how the state moves (for example how energy changes)
y     = g(x).T @ grad_H(x)                              # what the port delivers

You write	It means	Water tank example
H(x)	Total stored energy [J]	0.5 * rho * g * A * h**2
grad_H(x)	Gradient of that energy — the effort variable: pressure, voltage, force, temperature	Pressure at the base
J(x)	Power routed between internal stores, losslessly. Must satisfy J = -J.T	Zero — only one store
R(x)	Dissipation to heat. Must be positive semidefinite	Outlet orifice loss
g(x)	The ports — where power crosses the boundary	Inlet pipe

u is the port input (a flow: m³/s, A, N) and y the port output (an effort: Pa, V, m/s). Their product y.T @ u is power in watts.

Applications

	What J, R, H, g already are to you
Mechanical	A bond graph. J is the junction structure with its transformers and gyrators, R the R-elements, H the C- and I-elements
Electrical	An equivalent circuit. grad_H are node voltages and branch currents, J the lossless interconnection, R the resistive network
Chemical / process	An energy balance on a control volume, with internal exchange separated from irreversible loss
Control	A dissipative system in the sense of Willems. H is the storage function, y.T @ u the supply rate

The library only asks you to write the four parts down separately instead of collapsing them into a single right-hand side.

A complete model

import numpy as np
from otwin.model import PortHamiltonianSystem

A, a, c_d, rho, g_acc = 1.0, 0.1, 0.6, 1000.0, 9.81   # area, orifice, Cd, density, gravity

tank = PortHamiltonianSystem(
    H      = lambda x: 0.5 * rho * g_acc * A * float(x[0])**2,       # energy of the column
    grad_H = lambda x: np.array([rho * g_acc * A * x[0]]),           # pressure at the base [Pa]
    J      = lambda x: np.zeros((1, 1)),                             # one store, nothing circulates
    R      = lambda x: np.array([[c_d * a * np.sqrt(2 * g_acc / max(float(x[0]), 1e-9))
                                  / (rho * g_acc * A**2)]]),         # Torricelli outflow
    g      = lambda x: np.array([[1.0]]),                            # inlet pipe
    n_states=1, n_inputs=1,
)

for name, (ok, margin) in tank.check_structure(np.array([2.0])).items():
    print(f"{name:8s} {'pass' if ok else 'FAIL'}   margin {margin:.3e}")

J_skew   pass   margin 0.000e+00
R_psd    pass   margin 1.916e-05

check_structure is worth running every time you write a model. It catches sign errors and mis-transposed matrices immediately, before they turn into a plausible-looking wrong answer.

This model ships as otwin.model.water_tank. Also in the library you can find other examples: mass_spring_damper, dc_motor, pumped_hydro and heat_exchanger.

The port as feedback

A converter holding constant power, a thermostat, a droop-controlled inverter, a level valve — in all of them u depends on the state. If you pass a callable, the control law is evaluated at the step midpoint inside the implicit solve, which keeps the discrete power balance intact:

import numpy as np
from otwin.model import water_tank, integrate_phs

tank = water_tank()
t    = np.linspace(0, 600, 601)          # 10 minutes

# Inlet valve on level feedback: u is not a schedule, it is a control law.
res = integrate_phs(tank, np.array([2.0]), t,
                    u=lambda t, x: np.array([max(0.0, 5.0 * (1.5 - float(x[0])))]))

h, q = np.array(res["x"])[:, 0], np.array(res["u"])[:, 0]
print(f"level  {h[0]:.3f} m -> {h[-1]:.3f} m")
print(f"inflow {q[0]:.3f} -> {q[-1]:.3f} m3/s   (outflow at that level: "
      f"{0.6 * 0.1 * np.sqrt(2 * 9.81 * h[-1]):.3f} m3/s)")

level  2.000 m -> 1.436 m
inflow 0.000 -> 0.319 m3/s   (outflow at that level: 0.319 m3/s)

The tank settles below its 1.5 m setpoint, because proportional-only control leaves an offset. res["u"] is what the port actually did. with a governing law there is no schedule to read back.

Processes that produce entropy

A chemical reaction, heat conduction, a heat exchanger — any irreversible process needs more than reversible-plus-dissipative. IrreversiblePHS adds an entropy term to Port Hamiltonian Systems with sigma >= 0 enforced structurally; IrreversiblePHS.from_modulated(...) gives the Ramírez–Maschke–Sbarbaro form that most chemical reactor and heat exchanger models fall out. The IrreversiblePHS checked on every call the second law. otwin.model.heat_exchanger is a worked instance, with effectiveness_ntu for its steady-state check.

Hybrid Digital Twins (grey-box)

This is the most common case we can find. Combine all governing equations can be mathematically tricky and computationally unfeasible. These limitation can be managed with an hybrid approach combining basic governing equations plus observed data.

Hybrid Digital Twins are the most useful models in practice. Keep the analytic model as a prior and learn the residual, so the mean stays physically consistent and the correction carries calibrated uncertainty:

import numpy as np
from otwin.forecast import GPPHS

# The plant has quadratic drag. The analytic model we trust only knows linear damping,
# so the physics is right in structure and wrong in the loss term.
def truth(x, u):
    return np.array([x[1], -2.0*x[0] - 0.9*x[1]*abs(x[1])])

def phs_prior(x, u):
    return np.array([x[1], -2.0*x[0] - 0.3*x[1]])

rng  = np.random.default_rng(0)
X    = rng.uniform(-1.2, 1.2, size=(120, 2))
dXdt = np.array([truth(x, None) for x in X])

np.random.seed(0)   # GPPHS does not yet seed its own hyperparameter restarts
hybrid = GPPHS(n_states=2, prior_dynamics=phs_prior).fit(X, dXdt)

test  = rng.uniform(-1.0, 1.0, size=(40, 2))
exact = np.array([truth(x, None) for x in test])
mean, std = hybrid.predict(test, return_std=True)

err_prior  = np.sqrt(np.mean((np.array([phs_prior(x, None) for x in test]) - exact)**2))
err_hybrid = np.sqrt(np.mean((mean - exact)**2))
far        = hybrid.predict(np.array([[3.0, 3.0]]), return_std=True)[1].mean()

print(f"prior alone   RMSE on dx/dt : {err_prior:.4f}")
print(f"prior + GP    RMSE on dx/dt : {err_hybrid:.4f}")
print(f"GP std inside the fitted range : {std.mean():.4f}")
print(f"GP std far outside it          : {far:.4f}")

prior alone   RMSE on dx/dt : 0.1731
prior + GP    RMSE on dx/dt : 0.0055
GP std inside the fitted range : 0.0056
GP std far outside it          : 0.4593

Thirty times closer on the derivative, and — the part that matters operationally — the predictive standard deviation is two orders of magnitude larger at a state the GP has never seen. The correction knows when it is extrapolating even though the prior does not.

You can find a specific repository for modelling a Li-ion capacity in the repository [https://github.com/otwin-core/otwin-hybrid.git]

Uncertainty management

A measured orifice, a tolerance band on a capacitance, a coefficient quoted to two figures. Propagate it by making the spread the model:

import numpy as np
from otwin.model import water_tank
from otwin.forecast import Ensemble

# Parameter uncertainty, not measurement noise: the orifice was measured to +/-10 %.
members = [water_tank(a=a) for a in (0.090, 0.095, 0.100, 0.105, 0.110)]
ens     = Ensemble(members)

t, u  = np.linspace(0, 5, 101), np.zeros((101, 1))
band  = ens.forecast_interval(np.array([2.0]), t, u, level=0.90)
plant = water_tank(a=0.102).forecast(np.array([2.0]), t, u)["x"]   # the real orifice

for k in (0, 50, 100):
    print(f"t={t[k]:4.2f} s   band {band['lower'][k,0]:.3f}..{band['upper'][k,0]:.3f} m"
          f"   plant {plant[k,0]:.3f} m")
print(f"plant inside the band on {np.mean((plant >= band['lower']) & (plant <= band['upper'])):.0%} of steps")

t=0.00 s   band 2.000..2.000 m   plant 2.000 m
t=2.50 s   band 1.107..1.236 m   plant 1.156 m
t=5.00 s   band 0.476..0.656 m   plant 0.542 m
plant inside the band on 100% of steps

An ensemble of identical deterministic members has zero spread. That is correct, not a bug — the members must genuinely differ for the spread to mean anything.

The four steps pipeline

1. Estimating state from measurements

Estimator	Use it when
ExtendedKalmanFilter	The standard case: nonlinear model, Gaussian-ish noise
MovingHorizonEstimator	The state has physical bounds. A state of charge is not allowed to be 1.05
EnergyConsistentObserver	The correction itself must respect the energy balance

The moving-horizon estimator accepts box constraints on the state, and on the reference case it is also 39 % more accurate for it. The energy-consistent observer limits any correction so it cannot increase stored energy beyond what the ports supplied; the trade-off is in the docstring — with ports closed the allowed increase is zero, so a correction that merely reflects an under-energetic prior is rejected too.

2. Quantifying uncertainty

An interval has meaning if it's coverage has been measured: a stated 90 % interval should contain the truth about 90 % of the time.

import numpy as np
from otwin.forecast import rolling_origin_residuals, horizon_conformal

rng      = np.random.default_rng(0)
cycles   = np.arange(300)
capacity = 1.0 - 2.6e-4*cycles - 4.0e-3*np.sqrt(cycles) + rng.normal(0, 1.5e-3, 300)

class FadeLaw:
    """Fits C = C0 - a*n - b*sqrt(n) to the history, then extrapolates it."""
    def forecast(self, history, horizon):
        h = np.asarray(history, float).ravel()
        n = np.arange(len(h))
        coef, *_ = np.linalg.lstsq(np.column_stack([np.ones_like(n), n, np.sqrt(n)]), h, rcond=None)
        f = np.arange(len(h), len(h) + horizon)
        return (np.column_stack([np.ones_like(f), f, np.sqrt(f)]) @ coef).reshape(-1, 1)

train = capacity[:240]
def refit_forecast(history, horizon):
    return FadeLaw().forecast(history, horizon).ravel()

# Refit at earlier origins and keep genuine h-step-ahead errors -- not in-sample residuals.
residuals, horizons = rolling_origin_residuals(refit_forecast, train, step=5, max_horizon=60)
band = horizon_conformal(residuals, horizons, level=0.90, max_horizon=60)

lower, upper = band.apply(refit_forecast(train, 60))
truth = capacity[240:300]                      # the 60 cycles the band never saw

print(f"{residuals.size} residuals over horizons 1..{horizons.max()}")
print(f"half-width {(upper[0]-lower[0])/2:.4f} at h=1, {(upper[-1]-lower[-1])/2:.4f} at h=60")
print(f"measured coverage: {np.mean((truth >= lower) & (truth <= upper)):.0%}  (target 90%)")

1590 residuals over horizons 1..60
half-width 0.0023 at h=1, 0.0030 at h=60
measured coverage: 90%  (target 90%)

3. Validating a forecast

A model is not validated until its forecasts have been compared, out of sample, against a reference that is hard to beat. Same series and same FadeLaw as above:

from otwin.forecast import evaluate

print(evaluate(FadeLaw(), capacity.reshape(-1, 1),
               protocol="rolling_origin", n_folds=5, horizon=30))

EvalReport (rolling_origin, 5 folds)
────────────────────────────────────────────────────────────────
Skill Score (vs best baseline): 0.77 (77% better)
Baseline: persistence

Point Metrics:
  RMSE      0.0017 (baseline: 0.0074)
  MAE       0.0014 (baseline: 0.0064)
  NRMSE     0.1141
  MASE      0.8292
  THEIL_U   0.2395
────────────────────────────────────────────────────────────────

Three conventions are built into that one call, because each corrects a mistake that is easy to make and hard to notice:

• Partitions are out-of-sample by default. A random split on a time series trains on Tuesday and Thursday to predict Wednesday, which measures interpolation. random_split exists, warns loudly, and marks the report.
• A reference forecaster is required, and the report leads with the skill score. Model error divided by reference error, against the hardest of persistence, drift, mean and seasonal-naive. R² is available but never shown first: on a trending series it commonly reads above 0.95 for a model that loses to repeating the last observed value.
• The model is given history and a horizon, never the test values. If you supply exogenous drivers with exog=, every column is checked against the target first, and the report records that drivers were used — a skill score computed with the future of the drivers in hand is not comparable with one computed without.

4. Recording where the model is valid

The manifest carries which structure you chose, which parameters were estimated, how the model was validated and how the band was calibrated. The envelope turns that record into an answer or a refusal.

from otwin.advise import Envelope
from otwin.interfaces import TwinManifest, Provenance

twin = TwinManifest(
    name="cell-A3",
    model_class="empirical_law",          # one of: port_hamiltonian, irreversible_phs,
    model_kind="capacity_fade",           #         empirical_law, learned_phs, composite
    n_states=1, n_inputs=1,
    parameters={"C0": 0.9993, "a": 2.61e-4, "b": 4.02e-3},
    estimated=("C0", "a", "b"),           # three parameters came from data, so: grey box
    provenance=Provenance(created="2026-08-22", otwin_version="0.3.0", seed=0),
    validation=TwinManifest.validated_by("rolling_origin", rmse=0.0017, skill_score=0.77),
    calibration=TwinManifest.calibrated_by("horizon_conformal", empirical_coverage=0.90,
                                           level=0.90, max_horizon=60),
)
print("white box?", twin.is_white_box, " validated?", twin.is_validated)

envelope = Envelope(state_bounds=[(0.60, 1.00)], max_horizon=60)
print(envelope.check(state=[0.83], horizon=30,  manifest=twin).explain())
print(envelope.check(state=[0.83], horizon=180, manifest=twin).explain())

white box? False  validated? True
inside the validated envelope (horizon 30 <= 60; operating point inside the identified range;
validated, leakage-free)

outside the validated envelope:
  - horizon: beyond the validated forecast horizon (asked for 180, validated to 60)

This is a refusal, not a failure. The twin has not been shown to answer this question, and
returning a number anyway would hide that.

Same discipline as stating the calibration range of an instrument: a reading outside the calibrated range is reported as such, not returned as a number. is_validated is deliberately strict — leakage_free must be the boolean True, not merely truthy, because manifests arrive from MATLAB and Julia where a boolean can round-trip as 1 or as the string "false".

The pipeline, end to end

The modules follow the six data-processing blocks of ISO 13374, so the layout matches the reference architecture used in condition-monitoring practice.

Stage	Module	What happens
Data acquisition	otwin.io	Read the device over SunSpec or Modbus; normalise units
Data manipulation	otwin.signal	Resample to a uniform grid, find gaps, report coverage
State detection	otwin.estimate	Correct model state from measurements
Health assessment	otwin.model	Simulate the physical, learned or empirical model
Prognostic assessment	otwin.forecast	Forecast, score against a reference, calibrate the band
Advisory generation	otwin.advise	Report whether the request is inside the validated envelope

The otwin project

Repository	What it is
otwin	The library. Start here
otwin-spec	The specification and its conformance suite — reference cases with closed-form answers, used to verify that any implementation is correct, in any language
otwin-hybrid	A worked case in Python, Julia and R: predicting end of life of a lithium-ion cell from the first 40 % of its life. Opens in Colab in one click
otwin-systems	The growing catalogue of physical models, each shipped with a closed-form result it must reproduce

Julia and MATLAB implementations are open for contributors — otwin-spec tests over a subprocess interface, so a second-language implementation has an objective completion criterion: pass the suite unmodified. See CONTRIBUTING.md for that and for a list of self-contained project topics.

Issues, contribution and citation

• Questions and bugs — open an issue
• Contributing — CONTRIBUTING.md. The catalogue of physical models is the easiest place to start, and each contribution is a system you already know plus one closed-form result it must reproduce
• Citing — each repository carries a CITATION.cff

References

• van der Schaft, A. & Jeltsema, D. (2014). Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends in Systems and Control.
• Willems, J. C. (1972). Dissipative dynamical systems. Arch. Rational Mech. Anal. 45(5).
• Karnopp, D., Margolis, D. & Rosenberg, R. System Dynamics: Modeling, Simulation, and Control of Mechatronic Systems. Wiley.
• Ramírez, H., Maschke, B. & Sbarbaro, D. (2013). Irreversible port-Hamiltonian systems. Chemical Engineering Science 89.
• Greydanus, S., Dzamba, M. & Yosinski, J. (2019). Hamiltonian Neural Networks. NeurIPS 32.
• Rasmussen, C. E. & Williams, C. K. I. (2006). Gaussian Processes for Machine Learning. MIT Press.
• Vovk, V., Gammerman, A. & Shafer, G. (2005). Algorithmic Learning in a Random World. Springer.
• Gneiting, T. & Raftery, A. E. (2007). Strictly proper scoring rules, prediction, and estimation. JASA 102(477).
• ISO 13374 — Condition monitoring and diagnostics of machines.
• ISO 13381-1:2015 — Condition monitoring and diagnostics of machines: prognostics.