Online parameter identification for time-dependent PDEs via model reference adaptation.

How it works · Installation · Quick start · Examples · Verification · Citation

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Data of a state arrive over time; a parameter of the equation behind them is unknown. MRAS runs a copy of the model alongside the data, nudged towards them, and lets the mismatch drive an estimate of the parameter, step by step as the data come in. This repository turns the method into a toolbox: you declare the physics, it builds the forward model, the estimator and its solves, and checks the theory's assumptions for your problem.

_{Vector Burgers flow up a channel with three barriers. Left: the flow speed the data show, on a logarithmic scale. Middle: the viscosity MRAS reconstructs from those data as they arrive. Right: the true viscosity. Noise-free data, t = 0 to 25, on a mesh of width 0.025.}

How it works

For D_t u + f(q, u) = g with data z ≈ u†, the model reference adaptive system of Kaltenbacher & Nguyen (2021) evolves a state u and an estimate q together:

$$ \begin{aligned} D_t q + \sigma\big[D_t z + f(q,z) - g\big] - f_q'(\tilde q,z)^*(u-z) &= 0,\\ D_t u + f(q,z) + \mathcal{C}(u-z) &= g, \end{aligned} $$

with σ = 1 for f nonlinear in q, σ = 0 allowed for f affine in q, and a tracking operator C that pulls u towards the data. Under the assumptions (A1)–(A3) of Kaltenbach, Aarset & Nguyen (2026) with σ = 1, or persistently exciting data for f affine in q, q → q† and u → u†.

flowchart LR
    Z(["data z(t), arriving"]) --> S["state law<br/>D_t u + f(q, z) + C(u − z) = g"]
    Z --> P["parameter law<br/>D_t q = f_q′(q̃, z)* (u − z) − σ[…]"]
    S -- "misfit u − z" --> P
    P -- "estimate q" --> S
    P --> Q(["q(t) → q†"])

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Highlights

• Physics-driven via finite elements. A problem declares its model as weak-form densities; the driver builds the data model and executes the MRAS step
• Handles real data. MRAS is designed to handle real-world, noisy time series data, automatically regularising; it can also generate its own data for synthetic benchmarks
• Theory checked, not assumed. Every problem verifies the theoretical requirements of MRAS, reporting what can and cannot be guaranteed More below.
• Watch it run. A live seven-panel view in the notebook, movies rendered from HDF5 checkpoints, and exact restarts from them.

Installation

python3 -m venv .venv
.venv/bin/pip install -e ".[notebook,dev]"

ngsplot and ngsrng are installed from GitHub, at the latest commit of each. Movies need ffmpeg. The Darcy example reads one PDEBench file:

mkdir -p data && curl -L -o data/2D_DarcyFlow_beta100.0_Train.hdf5 \
  https://darus.uni-stuttgart.de/api/access/datafile/133221   # md5 7c09f3b1bb097737d3fd52ffb0c6f1c8

Quick start

.venv/bin/python -m mras.live burgers_2d --quick --noise-levels 0 0.05   # one example, small
.venv/bin/python scripts/run_examples.py burgers_2d                      # full size, with movie

from mras.live import build

prob = build("arrhenius", h=0.5, tend=10.0)   # the example's defaults, overridden
for r in prob.run(noise_levels=[0.0, 0.05]):
    print(r.noise_level, r.final_error_q)

Or open notebooks/mras_live.ipynb and watch an estimate converge.

Examples

example	equation, parameter q
darcy	time-dependent Darcy flow, permeability from PDEBench	AMMC §4
fisher	Fisher-type reaction–diffusion, diffusivity	AMMC Figs 5–6
potential	diffusion with a potential nonlinear in q	AMMC §5.2
allen_cahn	modified Allen–Cahn, nonlinear in u and q	AMMC §5.3
arrhenius, arrhenius_linear, arrhenius_sigma1	reaction rate q(x) r(u): every assumption verifiable
burgers_1d, burgers_1d_dark	scalar Burgers in a perforated pipe, diffusivity
burgers_2d	vector Burgers up a channel with barriers, viscosity (above)
blank_reaction	the template: copy examples/blank_problem.py

Your own problem

Copy examples/blank_problem.py and fill in its four marked blocks. A problem declares its physics and nothing else:

hook	declares
model(u, w), model_boundary(u, w)	the q-free part ⟨f(0, u), w⟩
coupling(q, u, w), coupling_adjoint(q, u, w)	the q-dependent part and its derivative's adjoint (required)
source(t), dirichlet(t), u0, q_true	forcing, boundary data, initial state, truth (for synthetic data)
C_scale, tracking_boundary(u, w)	the tracking operator C, −Δ by default
sigma; optionally L, c_coe()	the MRAS variant; analytic claims the checklist tests

Add a PROBLEM = MyProblem marker and a DEFAULTS dict, and python -m mras.live <name> runs it. The constructor names anything missing.

Verification

mras/verify.py asks every problem the same questions, at its audit discretization: the adjoint pairing of the assembled step, (A1)–(A3), persistent excitation of the data, the convergence statement for the declared σ, descent and reproducibility. Each answer is ok, fail, estimated, deviation or unverifiable, never a silent skip.

flowchart LR
    H["declared hooks<br/>and data"] --> U["universal checks<br/>refute or estimate"]
    U --> S{"structure<br/>detected"}
    S -- "f affine in q, Dirichlet part, …" --> X["exact constants"]
    D["declared bounds<br/>L, c_coe()"] --> C["confirmed,<br/>unless refuted"]
    U --> C

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.venv/bin/python scripts/checklist.py       # every example -> results/checklist.md

The tiers and the theory behind each check are set out in the docstring of mras/verify.py.

Reproducing results

	command
AMMC Fig 4, Table 6 (Darcy)	scripts/darcy_reproduce.py, then scripts/make_figures.py --problem darcy
AMMC Table 7, Figs 5–8	scripts/reproduce.py --problem fisher / --problem potential, then make_figures.py
AMMC Table 8, Figs 9–11	scripts/reproduce.py --problem allen_cahn, then make_figures.py
every example, every noise level	scripts/run_examples.py (resumable; --resume, --movie-from)
the animation above	scripts/readme_media.py

The paper's reproduction scripts take --micro for a seconds-long wiring check; their full-size runs take hours per noise level.

Repository layout

mras/        the toolbox: scheme, forward model, solvers, noise and smoothing,
             live view, movies, checkpoints, theory checklist, excitation maps
examples/    one file per problem, plus the blank template
configs/     YAML run configurations
scripts/     runners, paper reproductions, checklist, diagnostics
notebooks/   the live notebook
tests/       pytest suite (.venv/bin/pytest -q)
docs/media/  the animation above

Citation

If you use MRASLive, please cite the software (CITATION.cff) and the papers it implements.

@software{MRASLive,
  author  = {Kaltenbach, Benedikt and Aarset, Christian and Nguyen, Tram Thi Ngoc},
  title   = {{MRASLive}: online parameter identification for time-dependent {PDEs} by model reference adaptation},
  year    = {2026},
  version = {0.1.0},
  url     = {https://github.com/CAarset/MRASLive},
  license = {MIT}
}

@article{KaltenbachAarsetNguyen2026,
  author  = {Kaltenbach, Benedikt and Aarset, Christian and Nguyen, Tram Thi Ngoc},
  title   = {Data assimilation via model reference adaptation for linear and nonlinear dynamical systems},
  journal = {Applied Mathematics for Modern Challenges},
  volume  = {8},
  pages   = {80--112},
  year    = {2026},
  doi     = {10.3934/ammc.2026005}
}

@article{KaltenbacherNguyen2021,
  author  = {Kaltenbacher, Barbara and Nguyen, Tram Thi Ngoc},
  title   = {A model reference adaptive system approach for nonlinear online parameter identification},
  journal = {Inverse Problems},
  volume  = {37},
  number  = {5},
  pages   = {055006},
  year    = {2021}
}

License

MIT, for every version of this repository, including its earliest commits. The code it grew from, the authors' earlier MRAS code, was first released under CC BY 4.0; its copyright holders, the authors named in LICENSE, release it as contained here under the MIT License as well.