logo Insalogo Insa

Variational Data Assimilation & Model Learning


Variational Data Assimilation


Examples of inverse problems : least-squares (non-linear), optimal command, parameter identification, data assimilation.

Similarities with Artificial Neural Networks.

Optimal control:

            ODE Linear-Quadratic case, maximum principle, Hamiltonian.

            PDE non-linear, gradient computation via the adjoint equations, optimality system, Lagrangian.

Variational Data Assimilation.

                 Cost function, optimisation, regularisations.

Links between VDA, BLUE, sequential methods, Bayesian approach.


Model learning


Learning a model (ODE system or scalar PDEs) from large datasets.


Few programming practicals in Python: inverse problems  based on linear and non-linear advection-diffusion models.





Variational Data Assimilation (VDA)


50% of classes are on flipped mode. Lectures – exercises – Python codes.


A complete Moodle page is available with:

-        A more than complete course manuscript (140 pages with “to go further sections”),

-       exercises with their corrections,

-       programming practical(s) (lab tutorials) with Python codes.


The marked programming practical consists to solve a challenging inverse problem arising in Earth Sciences (spatial hydrology). Programming in Python (with Fenics library).


At the end of this module, the student will have understood and be able to explain (main concepts) :


Variational Data Assimilation


-  Fuse at best a PDE model with datasets.

- The optimal control of dynamic systems (ODE) and PDE models.

- To compute a gradient via the adjoint method.

- The basic principles of Automatic Differenciation.

- Algorithms of parameters identification, model calibration, Variational Data Assimilation (VDA).

- Introduce prior information via covariances matrix

- Links between VDA, BLUE, Kalman filtering and Bayesian approach.


Model learning


- Learning a model, ODE or (scalar)  PDE from datasets and an a-priori given dictionary.



The student will be able to :


Set up the equations and the complete modeling chain to perform parameters identification / model calibration / Variational Data Assimilation for PDE models.


Identify physical based model terms (e.g. ODE terms) from datasets.

Needed prerequisite

Variational Data Assimilation :


PDE (& ODE) models, differential calculus, optimisation, basis of functional analysis, numerical schemes, programming.

Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...


- "Variational data assimilation & Model Learning " by J. Monnier. INSA - Univ Toulouse. Course manuscript 144 pages



- Asch, M., Bocquet, M. and Nodet, M., 2016. Data assimilation: methods, algorithms, and applications. Society for Industrial and Applied Mathematics.


- “Computational Methods for Data Evaluation and Assimilation”.Dan Gabriel Cacuci, Ionel Michael Navon, Mihaela Ionescu-Bujor. Chapman and Hall, 2013.


- “Data Assimilation for Scientists and Engineers” : a Toulouse University pedagogical project. O. Thual, S. Gratton, G. Monnier, O. Pannekoucke. http://pedagotech.inp-toulouse.fr/130107/