Course & textbook · INSA Toulouse, Applied Mathematics & Master Mod'IA - Scientific ML

Data Assimilation, Adjoint Methods and Learning

Inverse Problems for Physics-Based Models, from Optimal Control to Hybrid Machine Learning

How to fuse physics-based models (ODEs, PDEs) with observations? The course covers the estimation of model parameters, initial or boundary conditions and forcing terms; the computation of gradients in large dimension by the adjoint method; and the many bridges between data assimilation and machine learning.

Textbook (PDF, HAL) Moodle course page How to cite

Moodle, open access:

The course in three pictures

Diagram of the VDA principle: input parameters feed a model and its computational code; the model output is compared to observations through a cost function, and the discrepancy is fed back to the uncertain inputs.
1. The principle: from direct modeling to model calibration. A physics-based model (ODE or PDE, here the 2D shallow-water equations) depends on uncertain inputs: empirical coefficients, initial and boundary conditions. Its output is compared to observations (satellite, airborne or in-situ) through a cost function, which variational data assimilation (VDA) minimizes with respect to the inputs. The key ingredient is the gradient with respect to possibly millions of inputs: the adjoint model provides it for about the cost of one extra model run (Parts II–IV).
Lorenz-63 twin experiment: the free run diverges from the truth while the EnKF and 4D-Var analyses track it; bottom panel shows RMSE below the observation error.
2. Variational vs sequential DA on a chaotic system. Lorenz-63 twin experiment: noisy observations (red dots) are generated from a known true trajectory (black). Started from a slightly wrong initial state, the model alone (orange) soon loses track of the truth — the hallmark of chaos. An ensemble Kalman filter and a cycled 4D-Var (adjoint gradient, L-BFGS) both follow it, with errors below the observation noise (bottom), since the dynamics propagates past information. 4D-Var is the most accurate; the EnKF also estimates its own uncertainty (Chapters 5, 13, 15).
Hybrid model vs neural ODE on a damped oscillator: (a) fit and extrapolation in time, (b) new initial condition, (c) learned friction law versus the true one.
3. Physics + learning: hybrid model vs pure neural network. A damped oscillator with unknown friction is observed on a short window (grey band). Two models are trained by the same adjoint-based gradient: a neural ODE, whose whole dynamics is a network (black box), and a hybrid model which keeps the known physics and learns only the friction (grey box). (a) Both fit the data, but the neural ODE learned the trajectory, not the law: its extrapolation error is 10 times larger. (b) From a new initial state, both degrade; (c) the learned friction is accurate only over the velocities seen in the data. A two-parameter physical law extrapolates more robustly. Lessons of Chapter 7: physics improves generalization, learned terms are reliable only where data exist.

Contents

Part IInverse problems and learning from data

Ill-posedness; least squares, SVD, Tikhonov regularization, parameter choice; neural networks as parametric models; a running example: river bathymetry.

Part IIVariational DA and the adjoint method

Overview of DA methods; VDA in finite dimension; optimal control of ODEs; PINNs, and PINNs vs VDA.

Part IIIGradients: computation & validation

Backpropagation is the adjoint method; automatic differentiation; validating gradients and adjoint codes.

Part IVPDE models and 4D-Var

Optimal control of stationary PDEs; VDA for time-dependent PDEs: 4D-Var, incremental 4D-Var.

Part VStatistical estimation & sequential DA

Bayesian inversion, MAP, sampling; BLUE, Kalman filters, ensemble methods; background error covariances.

Part VIPerspectives

Hybridizing data assimilation and machine learning. Appendices: differential calculus, optimization, statistics, solutions of the exercises.

Each chapter comes with exercises and Python codes; "learning corners" connect the topic to machine learning.

Practicals

River bathymetry from water-surface data

A real-world inverse problem in spatial hydrology, solved by VDA (Python, FEniCS finite elements).

Optimal control, adjoint and learning

Controlling a dynamical system, computing and validating the adjoint, and comparing with learning approaches (Python).

Statements, codes and course sessions are on the Moodle page. For research use in hydrology, see the open-source DassHydro platform.

How to cite

Jérôme Monnier. Data Assimilation, Adjoint Methods and Learning. Inverse Problems for Physics-Based Models, from Optimal Control to Hybrid Machine Learning. Lecture notes, INSA Toulouse, 2026. ⟨hal-03040047⟩

BibTeX 
@misc{Monnier-DA-course,
  author       = {Monnier, J{\'e}r{\^o}me},
  title        = {Data Assimilation, Adjoint Methods and Learning.
                  Inverse Problems for Physics-Based Models,
                  from Optimal Control to Hybrid Machine Learning},
  howpublished = {Lecture notes, INSA Toulouse},
  year         = {2026},
  note         = {HAL: hal-03040047},
  url          = {https://hal.science/hal-03040047}
}
HAL record (all versions)

Prerequisites

Basic approximation methods, differential calculus, numerical optimization, classical PDE models (weak forms are a plus), numerical schemes (finite differences; finite elements are a plus), artificial neural networks, Python programming.

The textbook is a living document (October 2026 version), revised as the course progresses. Questions and feedback are welcome by e-mail.