physics-informed neural networks

The Navier-Stokes equations are a collection of partial differential equations that describe fluid motion in liquids and gases, and provide a mathematical framework for modeling fluid behavior in various scenarios. Solving them correctly reveals information about fluid behavior such as turbulence, laminar flow, and vortex formation. In practical applications, numerical methods such as finite difference, finite element, and computational fluid dynamics (CFD) are frequently used to approximate Navier-Stokes equation solutions. Solving the Navier-Stokes equations is a challenging task due to their nonlinearity and complexity, The main idea behind PINNs is to incorporate a physical system's governing equations into the neural network's training process. In this study, the Navier-Stokes equation is solved using physicsinformed neural networks(PINN) to provide insight into the potential of PINNs in fluid dynamics for computationally less expensive calculations without compromising the accuracy using L-BFGS optimizer algorithm for computationally less expensive(CLE) calculations.

The measurement of black hole spin is considered one of the key problems in relativistic astrophysics. Existing methods, such as continuum fitting, X-ray reflection spectroscopy and quasi-periodic oscillation analysis, have systematic limitations in accuracy, interpretability and scalability. In this work, a hybrid approach is proposed in which theoretical models based on the Teukolsky formalism are integrated with Physics-Informed Neural Networks (PINNs). A PINN model is developed to solve the linearized spin problem in the scalar case, with physical constraints directly embedded into the training process. Annotated data are not required; instead, the model is trained using the differential operator and boundary conditions as supervision. It is demonstrated that the PINN converges reliably, with residual loss values below 10 and a root mean squared error (RMSE) of the order of10-6 (final ≈ 5.4 × 10-8). Benchmarking results indicate that the proposed method outperforms both classical and data-driven machine learning approaches in terms of AUC and sensitivity, while also exhibiting superior interpretability, generalizability and adherence to physical principles, with moderate computational cost. Potential extensions include integration with general relativistic magnetohydrodynamics (GRMHD) solvers and application to real observational data. These findings support the viability of physics-based machine learning as a robust framework for accurate and interpretable black hole spin estimation.

Surrogate modelling is increasingly used in engineering to improve computational efficiency in complex simulations. However, traditional data-driven surrogate models often face limitations in generalizability, physical consistency, and extrapolation-issues that are especially critical in safety-sensitive fields such as fire safety engineering (FSE). To address these concerns, physics-informed surrogate modelling (PISM) integrates physical laws into machine learning models, enhancing their accuracy, robustness, and interpretability. This systematic review synthesises existing applications of PISM in FSE, classifies the strategies used to embed physical knowledge, and outlines key research challenges. A comprehensive search was conducted across Google Scholar, ResearchGate, ScienceDirect, and arXiv up to May 2025, supported by backward and forward snowballing. Studies were screened against predefined criteria, and relevant data were analysed through narrative synthesis. A total of 100 studies were included, covering five core FSE domains: fire dynamics, wildfire behaviour, structural fire engineering, material response, and heat transfer. Four main strategies for embedding physics into machine learning were identified: feature engineering techniques (FETs), loss-constrained techniques (LCTs), architecture-constrained techniques (ACTs), and offline-constrained techniques (OCTs). While LCT and ACT offer strict enforcement of physical laws, hybrid approaches combining multiple strategies often produce better results. A stepwise framework is proposed to guide the development of PISM in FSE, aiming to balance computational efficiency with physical realism. Common challenges include handling nonlinear behaviour, improving data efficiency, quantifying uncertainty, and supporting multi-physics integration. Still, PISM shows strong potential to improve the reliability and transparency of machine learning in fire safety applications.

This paper presents a novel framework for a Theory of Everything (ToE) that unifies consciousness with non-commutative spacetime, superstring theory, loop quantum gravity (LQG), and holographic principles. By introducing a consciousness field, ψ C , we extend existing physical theories to incorporate quantum consciousness, exotic matter, negative mass, wormholes, gravitons, tachyons, anti-matter, non-linear time currents, dark matter, and extra dimensions. The objectives are to provide a mathematical foundation for consciousness as a fundamental entity, reconcile quantum mechanics with general relativity, and propose testable predictions. This work integrates with established theories such as string theory (1), LQG (3), and holography (5), while introducing novel concepts like exotic charges and non-linear time dynamics.

This paper presents a new approach to analyzing the controllability of fractional Volterra-Fredholm integro-differential equations with state-dependent delay, characterized by the Caputo fractional derivative and governed by a semigroup of compact and analytic operators. Controllability results are derived using Schauder's fixed point theorem, addressing the challenges of fractional dynamics and state-dependent delays. A key innovation in this study is the integration of theoretical analysis with advanced computational methods, specifically Physics-Informed neural networks, to approximate solutions and verify controllability conditions. To validate the theoretical findings, a detailed example is provided, along with numerical simulations that confirm the convergence of solutions. Graphical representations offer additional insights into the solution dynamics, enhancing the understanding of the system's behavior. By combining mathematical rigor with machine learning techniques, this work establishes a computational framework for tackling complex fractional systems, paving the way for further exploration of their controllability.

La thermodynamique est une expression de la physique à un niveau épistémique élevé. À ce titre, son potentiel en tant que biais inductif pour aider les procédures d'apprentissage automatique à obtenir des prédictions précises et crédibles a été récemment reconnu dans de nombreux domaines. La thermodynamique fournit des informations utiles dans le processus es phénomènes dissipatifs. Nous allons expliquer le lien entre les TINN (Réseaux de Neurones Informés par la Thermodynamique), le flot métriplectique et la thermodynamique des groupes de Lie et comment ces concepts cherchent collectivement à intégrer les principes de la thermodynamique et des symétries dans la modélisation mathématique, l'apprentissage automatique et les systèmes physiques.

Many insurance products are structured like options, offering risk management akin to financial derivatives. Accurate option pricing techniques are crucial for market stability and investor confidence. This study implements an approach to option pricing by implementing a physics-informed neural network (PINN) with an enhanced learning rate, transformer layer, and the combination of the two (combined PINN) to solve the Black-Scholes partial differential equation (PDE) for European and American options. Different activation functions (tanh, swish, GELU (Gaussian error linear unit), and SELU (scaled exponential linear unit)) are tested across models, with model performance evaluated using root mean square error (RMSE). The model successfully approximates the Black-Scholes solution well for European and American options. The best-performing model is the combined PINN for both European call and put with the best RMSE 0.00542 and 0.00126, respectively. The results show that the combined PINN model is better for solving the Black-Scholes PDE.

We conducted a comprehensive comparative study of numerical solvers for the generalized Korteweg-de Vries (gKdV) equation, focusing on classical Fourier-based Crank-Nicolson methods and physics-informed neural networks (PINNs). Our work benchmarks these approaches across nonlinear regimes-including the cubic case (ν = 3)-and diverse initial conditions such as solitons, smooth pulses, discontinuities, and noisy profiles. In addition to pure PINN and spectral models, we propose a novel hybrid PINN-spectral method incorporating a regularization term based on Fourier reference solutions, leading to improved accuracy and stability. Numerical experiments show that while spectral methods achieve superior efficiency in structured domains, PINNs provide flexible, mesh-free alternatives for data-driven and irregular setups. The hybrid model achieves lower relative L 2 error and better captures soliton interactions. Our results demonstrate the complementary strengths of spectral and machine learning methods for nonlinear dispersive PDEs.

We shall elucidate the foliation structures, namely, the 3-web and the bi-Lagrangian structure, that were jointly employed by the physicist and mathematician Jean-Marie Souriau in his Lie Groups Thermodynamics, extended to include dissipation models, and by Gérard Debreu, the Nobel Laureate in Economics, within the context of preferences and utility theory. Debreu examined the conditions under which a web is considered trivial, that is, whether there exists a change of coordinates rendering the web equivalent to a standard orthogonal grid, integrable into a potential function. He employed the Frobenius theorem and Pfaffian forms to investigate whether certain distributions, in the sense of foliations, satisfy the necessary conditions for integrability. Souriau developed a symplectic model of thermodynamics based on symplectic foliation, wherein dissipation dynamics are defined on a transverse Riemannian foliation, thereby inducing a web structure linked to a bi-Lagrangian manifold. A bi-Lagrangian manifold may be endowed with a metric 3-web structure via a Riemannian metric. For every 3-web, one can associate a canonical Chern connection, whose flatness guarantees additive separability. This connection, originally introduced by Hess for Lagrangian 2-webs, also known as bipolarised symplectic manifolds, is particularly employed in the study of bi-Lagrangian manifolds.

In this contribution the authors propose a hybrid Boundary Element Method-Physics Informed Neural Networks (BEM-PINN) approach, to be used for the resolution of partial differential equations arising when formulating boundary-value problems in electromagnetism. The approach retains both the advantages of integral methods (compact representation and no need to mesh large domains) and differential methods, where the term ''differential'' refers here to the Automatic Differentiation carried out during the training phase of the PINN. The method is easy to implement and adds an additional flexibility to purely PINN based solution methods. INDEX TERMS Boundary element method, physics informed neural networks, Laplace equation.

Jean-Marie Souriau's model, known as "Lie groups Thermodynamics," is a symplectic model of statistical mechanics. This framework integrates geometric methods into statistical mechanics, where Gibbs states of a system are represented as points in a symplectic manifold, and Lie group describes the symmetries of the system. We give an original interpretation of “Lie groups Thermodynamics” with a symplectic foliation generated by coadjoint orbit of the Lie group acting on the system, where the Entropy is characterized as an invariant Casimir Function. Transverse to this symplectic foliation, considered as level sets of Entropy, we can associate a Riemannian foliation as level set of Energy. These transverse foliations define a web structure. We explain dynamics along both transverse leaves by a metriplectic flow mixing Poisson bracket on symplectic leaves (level sets of Entropy), describing non-dissipative phenomenon by preserving Entropy, and metric bracket on Riemannian leaves (level sets of Energy), characterizing dissipative phenomenon by Entropy production. In the framework of Information Geometry for statistical manifolds, we can associate to the symplectic foliation the Fisher metric, and to the Riemannian foliation the dual of the Fisher metric (hessian of Entropy).
We develop this foliation model of thermodynamics, making the link with the notion of metriplectic flow, compatible with Onsager relations, describing the dynamics along each of the leaves. We show that to the symplectic foliation of Souriau, we can associate a transverse Riemannian foliation by the energy level sets, which describe the dissipative phenomena and the generation of Entropy of the 2nd principle of Sadi Carnot. We recall that the physicist Baptiste Coquinot has established the compatibilities of metriplectic flow with the Onsager relations that describe dissipative phenomena. We also recall that Günther Vojta had studied first Onsager relations within the framework of symplectic geometry. We conclude with physicist Herbert B. Callen's insight into thermodynamics as the Science of symmetry. We recall avenues of study opened by Callen to explore more deeply the role of symmetry in thermodynamics.

Thermodynamics understanding by Geometric model were initiated by all precursors Carnot, Gibbs, Duhem, Reeb, Carathéodory. It is only recently that Symplectic Foliation Model introduced in the domain of Geometric statistical Mechanics has opened the door for a solid  bedrock, giving a geometric definition of Entropy as invariant Casimir function on Symplectic leaves (coadjoint orbit of the Lie Group acting on the system and interpreted as level sets of Entropy).

As observed by Georges Reeb "Thermodynamics has long accustomed mathematical physics [see DUHEM P.] to the consideration of completely integrable Pfaff forms: the elementary heat dQ [notation of thermodynamicists] representing the elementary heat yielded in an infinitesimal reversible modification is such a completely integrable form. This point does not seem to have been explored since then." Notion of foliation in thermodynamics appears in  C. Carathéodory paper where horizontal curves roughly correspond to adiabatic processes, performed in the language of Carnot cycles.  The properties of the couple of Poisson manifolds was  previously explored by C. Carathéodory in 1935, under the name of “function groups, polar to each other”. This seminal work of C. Caratheodory leads to the concept of a Poisson structure which was first defined independently by Lichnerowicz and  Kirillov.

A symplectic foliation model of Thermodynamics has been defined by Jean-Marie Souriau based on "Lie Groups Thermodynamics" model. This model gives a cohomological characterization of Entropy, as an invariant Casimir function in coadjoint representation. The dual space of the Lie algebra foliates into coadjoint orbits identified with the Entropy level sets. In the framework of Thermodynamics, a symplectic bifoliation structure is associated to describe non-dissipative dynamics on symplectic leaves (on level sets of Entropy as constant Casimir function on each leaf), and transversal dissipative dynamics, given by Poisson transverse structure (Entropy production from leaf to leaf). The symplectic foliation orthogonal to the level sets of moment map is the foliation determined by hamiltonian vector fields generated by functions on dual Lie algebra. The orbits of a Hamiltonian action and the level sets of its moment map are polar to each other. The space of Casimir functions on a neighborhood of a point is isomorphic to the space of Casimirs for the transverse Poisson structure. Souriau’s model could be then interpreted by Libermann's foliations, clarified as dual to Poisson Gamma-structure of Haefliger, which is the maximum extension of the notion of moment in the sense of J.M. Souriau, as introduced by P. Molino, M. Condevaux and P. Dazord in papers of  “Séminaire Sud-Rhodanien de Géométrie ». The symplectic duality to a symplectically complete foliation, in the sense of Libermann, associates an orthogonal foliation. We conclude with link to Cartan foliation and Edmond Fedida works on Cartan's mobile frame-based foliation.

In the first part, we will present the theme "Statistical learning on Lie groups" [1,2] which extends statistics and machine learning to Lie groups based on the theory of representations and cohomology of Lie algebra. From the work of Jean-Marie Souriau on " Lie Groups Thermodynamics" [4] initiated within the framework of symplectic models of statistical mechanics, new geometric statistical tools have been developed to define probability densities (Gibbs density of Maximum Entropy) on Lie Groups or homogeneous manifolds for supervised methods, and the extension of the Fisher metric of Information Geometry for unsupervised methods (in metric spaces).

In a 2nd part, TINNs (Thermodynamics-Informed Neural Networks) [3,5] will be discussed for AI-augmented engineering applications. The geometric structures of TINNs are studied by their metriplectic flow (also called GENERIC flow) modeling non-dissipative dynamics (1st thermodynamic principle of energy conservation) and dissipative dynamics (2nd thermodynamic principle of entropy production). The Thermodynamics of Lie Groups of Souriau makes it possible to geometrically characterize the metriplectic flow by a structure of “webs” composed of symplectic foliations and transversely Riemannian foliations. From the symmetries of the problem, the coadjoint orbits of the Lie group generate via the moment map (geometrization of Noether's theorem) the symplectic foliation (defined as the level sets of entropy, where entropy is an invriant Casimir function on these symplectic leaves). The metric on symplectic leaves is given by the Fisher metric. The dynamics along these symplectic leaves, given by the Poisson bracket, characterizes the non-dissipative dynamics. The dissipative dynamics is then given by the transverse Poisson structure and a metric flow  bracket, with an evolution from leaf to leaf constrained by the production of entropy. The transverse foliation is a Riemannian foliation whose metric is given by the dual of Fisher metric (Hessian of Entropy).

These machine-learning tools on Lie groups and TINNs are addressed in two European projects: a European network COST CaLISTA [6] and the European Marie-Curie action MSCA CaLIGOLA [7].

References:

[1] Barbaresco, F. (2022) Symplectic theory of heat and information geometry, chapter 4, Handbook of Statistics, Volume 46, Pages 107-143, Elsevier, https://doi.org/10.1016/bs.host.2022.02.003 https://www.sciencedirect.com/science/article/abs/pii/S0169716122000062 

[2] Barbaresco, F. (2023). Symplectic Foliation Transverse Structure and Libermann Foliation of Heat Theory and Information Geometry. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14072. Springer, Cham. https://doi.org/10.1007/978-3-031-38299-4_17  ; https://link.springer.com/chapter/10.1007/978-3-031-38299-4_17 

[3] Barbaresco, F. (2022) Symplectic Foliation Structures of Non-Equilibrium Thermodynamics as Dissipation Model: Application to Metriplectic Nonlinear Lindblad Quantum Master Equation. Entropy 2022, 24, 1626. https://doi.org/10.3390/e24111626

[4] Souriau, J.M. (1969). Structure des systèmes dynamiques. Dunod. http://www.jmsouriau.com/structure_des_systemes_dynamiques.htm   

[5] Cueto E. and Chinesta F. (2022), Thermodynamics of Learning of Physical Phenomena, arXiv:2207.12749v1 [cs.LG] 26 Jul 2022

[6] European COST network CA21109 : CaLISTA  - Cartan geometry, Lie, Integrable Systems, quantum group Theories for Applications ; https://site.unibo.it/calista/en

[7] European HORIZON-MSCA-2021-SE-01-01 project CaLIGOLA - Cartan geometry, Lie and representation theory, Integrable Systems, quantum Groups and quantum computing towards the understanding of the geometry of deep Learning and its Applications; https://site.unibo.it/caligola/en

[8] Barbaresco, F.  (2021) Koszul lecture related to geometric and analytic mechanics, Souriau’s Lie group thermodynamics and information geometry. Info. Geo. 4, 245–262. https://doi.org/10.1007/s41884-020-00039-x

[9]  Barbaresco, F. (2021). Gaussian Distributions on the Space of Symmetric Positive Definite Matrices from Souriau’s Gibbs State for Siegel Domains by Coadjoint Orbit and Moment Map. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_28

[10] Barbaresco, F. (2021). Koszul Information Geometry, Liouville-Mineur Integrable Systems and Moser Isospectral Deformation Method for Hermitian Positive-Definite Matrices. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_39

[11] Barbaresco, F. (2021). Jean-Marie Souriau’s Symplectic Model of Statistical Physics: Seminal Papers on Lie Groups Thermodynamics - Quod Erat Demonstrandum. In: Barbaresco, F., Nielsen, F. (eds) Geometric Structures of Statistical Physics, Information Geometry, and Learning. SPIGL 2020. Springer Proceedings in Mathematics & Statistics, vol 361. Springer, Cham. https://doi.org/10.1007/978-3-030-77957-3_2

[12] 7th SEE Geometic Science of Information; https://conference-gsi.org/

This study explores a hybrid framework integrating machine learning techniques and symbolic regression via genetic programming for analyzing the nonlinear propagation of waves in arterial blood flow. We employ a mathematical framework to simulate viscoelastic arterial flow, incorporating assumptions of long wavelength and large Reynolds numbers. We used a fifth-order nonlinear evolutionary equation using reductive perturbation to represent the behavior of nonlinear waves in a viscoelastic tube, considering the tube wall's bending. We obtain solutions through physics-informed neural networks (PINNs) that optimizes via Bayesian hyperpa-rameter optimization across three distinct initial conditions. We found that physics-informed neural network-based models are proficient at predicting the solutions of higher-order nonlinear partial differential equations in the spatial-temporal domain [−1, 1]×[0, 2]. This is evidenced by graphical results and a residual validation showing a mean absolute residue error of O(10 −3). We thoroughly examine the impacts of various initial conditions. Furthermore, the three solutions are combined into a single model using the random forest machine learning algorithm, achieving an impressive accuracy of 99% on the testing dataset and compared with another model using an artificial neural network. Finally, the analytical form of the solutions is estimated using symbolic regression that provides interpretable models with mean square error of O(10 −3). These insights contribute to the interpretation of cardio-vascular parameters, potentially advancing machine learning applications within the medical domain.

Freeze casting, a manufacturing technique widely applied in biomedical fields for fabricating biomaterial scaffolds, poses challenges for predicting directional solidification due to its highly nonlinear behavior and complex interplay of process parameters. Conventional numerical methods, such as computational fluid dynamics (CFD), require adequate and accurate boundary condition knowledge, limiting their utility in real-world transient solidification applications due to technical limitations. In this study, we address this challenge by developing a physics-informed neural networks (PINNs) model to predict directional solidification in freeze-casting processes. The PINNs model integrates physical constraints with neural network predictions, requiring significantly fewer predetermined boundary conditions compared to CFD. Through a comparison with CFD simulations, the PINNs model demonstrates comparable accuracy in predicting temperature distribution and solidification patterns. This promising model achieves such a performance with only 5000 data points in space and time, equivalent to 250,000 timesteps, showcasing its ability to predict solidification dynamics with high accuracy. The study's major contributions lie in providing insights into solidification patterns during freeze-casting scaffold fabrication, facilitating the design of biomaterial scaffolds with finely tuned microstructures essential for various tissue engineering applications. Furthermore, the reduced computational demands of the PINNs model offer potential cost and time savings in scaffold fabrication, promising advancements in biomedical engineering research and development.

The significance of Ultimate Bond Stress-Slip (UBS-S) in reinforced Ultra-High Performance Concrete (UHPC) structures cannot be overstated, as it directly affects their load-carrying capacity, structural integrity, and long term performance. A comprehensive analysis of the UHPC-Parallel Micro Element System (UHPC-PMES),
including 144 specimens, evaluated the computational efficiency of the proposed UBS-S model. To this end, the most critical settings of steel bar diameter, concrete cover, bond length, and UHPC compressive strength (db, c, lb,f′UHPC) were directed to create this parametric research. Applying the hybrid approach of three different optimization
techniques, namely Physics-Informed Neural Networks (PINN), Genetic Algorithms (GA), and Multiple Linear Regression (MLR), this study predicted the UBS-S at the interface of UHPC and steel bars. It formulated the hyper-parameters effect values (a, m, β, G). The presented research used these algorithms to solve an inverse
problem in structural engineering. Comparing the results obtained from PINN, GA, and MLR demonstrated that machine learning techniques and the proposed PMES model could effectively and accurately investigate the ultimate bond stress-slip for reinforced UHPC structures.

Thermodynamics understanding by Geometric model were initiated by all precursors Carnot, Gibbs, Duhem, Reeb, Carathéodory. It is only recently that Symplectic Foliation Model introduced in the domain of Geometric statistical Mechanics has opened the door for a solid  bedrock, giving a geometric definition of Entropy as invariant Casimir function on Symplectic leaves (coadjoint orbit of the Lie Group acting on the system and interpreted as level sets of Entropy).

As observed by Georges Reeb "Thermodynamics has long accustomed mathematical physics [see DUHEM P.] to the consideration of completely integrable Pfaff forms: the elementary heat dQ [notation of thermodynamicists] representing the elementary heat yielded in an infinitesimal reversible modification is such a completely integrable form. This point does not seem to have been explored since then." Notion of foliation in thermodynamics appears in  C. Carathéodory paper where horizontal curves roughly correspond to adiabatic processes, performed in the language of Carnot cycles.  The properties of the couple of Poisson manifolds was  previously explored by C. Carathéodory in 1935, under the name of “function groups, polar to each other”. This seminal work of C. Caratheodory leads to the concept of a Poisson structure which was first defined independently by Lichnerowicz and  Kirillov.

A symplectic foliation model of Thermodynamics has been defined by Jean-Marie Souriau based on "Lie Groups Thermodynamics" model. This model gives a cohomological characterization of Entropy, as an invariant Casimir function in coadjoint representation. The dual space of the Lie algebra foliates into coadjoint orbits identified with the Entropy level sets. In the framework of Thermodynamics, a symplectic bifoliation structure is associated to describe non-dissipative dynamics on symplectic leaves (on level sets of Entropy as constant Casimir function on each leaf), and transversal dissipative dynamics, given by Poisson transverse structure (Entropy production from leaf to leaf). The symplectic foliation orthogonal to the level sets of moment map is the foliation determined by hamiltonian vector fields generated by functions on dual Lie algebra. The orbits of a Hamiltonian action and the level sets of its moment map are polar to each other. The space of Casimir functions on a neighborhood of a point is isomorphic to the space of Casimirs for the transverse Poisson structure. Souriau’s model could be then interpreted by Libermann's foliations, clarified as dual to Poisson Gamma-structure of Haefliger, which is the maximum extension of the notion of moment in the sense of J.M. Souriau, as introduced by P. Molino, M. Condevaux and P. Dazord in papers of  “Séminaire Sud-Rhodanien de Géométrie ». The symplectic duality to a symplectically complete foliation, in the sense of Libermann, associates an orthogonal foliation. We conclude with link to Cartan foliation and Edmond Fedida works on Cartan's mobile frame-based foliation.

In the first part, we will present the theme "Statistical learning on Lie groups" [1,2] which extends statistics and machine learning to Lie groups based on the theory of representations and cohomology of Lie algebra. From the work of Jean-Marie Souriau on " Lie Groups Thermodynamics" [4] initiated within the framework of symplectic models of statistical mechanics, new geometric statistical tools have been developed to define probability densities (Gibbs density of Maximum Entropy) on Lie Groups or homogeneous manifolds for supervised methods, and the extension of the Fisher metric of Information Geometry for unsupervised methods (in metric spaces).

In a 2nd part, TINNs (Thermodynamics-Informed Neural Networks) [3,5] will be discussed for AI-augmented engineering applications. The geometric structures of TINNs are studied by their metriplectic flow (also called GENERIC flow) modeling non-dissipative dynamics (1st thermodynamic principle of energy conservation) and dissipative dynamics (2nd thermodynamic principle of entropy production). The Thermodynamics of Lie Groups of Souriau makes it possible to geometrically characterize the metriplectic flow by a structure of “webs” composed of symplectic foliations and transversely Riemannian foliations. From the symmetries of the problem, the coadjoint orbits of the Lie group generate via the moment map (geometrization of Noether's theorem) the symplectic foliation (defined as the level sets of entropy, where entropy is an invriant Casimir function on these symplectic leaves). The metric on symplectic leaves is given by the Fisher metric. The dynamics along these symplectic leaves, given by the Poisson bracket, characterizes the non-dissipative dynamics. The dissipative dynamics is then given by the transverse Poisson structure and a metric flow  bracket, with an evolution from leaf to leaf constrained by the production of entropy. The transverse foliation is a Riemannian foliation whose metric is given by the dual of Fisher metric (Hessian of Entropy).

These machine-learning tools on Lie groups and TINNs are addressed in two European projects: a European network COST CaLISTA [6] and the European Marie-Curie action MSCA CaLIGOLA [7].
References:

[1] Barbaresco, F. (2022) Symplectic theory of heat and information geometry, chapter 4, Handbook of Statistics, Volume 46, Pages 107-143, Elsevier, https://doi.org/10.1016/bs.host.2022.02.003 https://www.sciencedirect.com/science/article/abs/pii/S0169716122000062

[2] Barbaresco, F. (2023). Symplectic Foliation Transverse Structure and Libermann Foliation of Heat Theory and Information Geometry. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14072. Springer, Cham. https://doi.org/10.1007/978-3-031-38299-4_17 ; https://link.springer.com/chapter/10.1007/978-3-031-38299-4_17 

[3] Barbaresco F. (2022). Symplectic Foliation Structures of Non-Equilibrium Thermodynamics as Dissipation Model: Application to Metriplectic Non-Linear Lindblad Quantum Master Equation, submitted to MDPI special Issue "Geometric Structure of Thermodynamics: Theory and Applications", 2022

[4] Souriau, J.M. (1969). Structure des systèmes dynamiques. Dunod. http://www.jmsouriau.com/structure_des_systemes_dynamiques.htm 

[5] Cueto E. and Chinesta F. (2022), Thermodynamics of Learning of Physical Phenomena, arXiv:2207.12749v1 [cs.LG] 26 Jul 2022

[6] European COST network CA21109 : CaLISTA  - Cartan geometry, Lie, Integrable Systems, quantum group Theories for Applications ; https://site.unibo.it/calista/en

[7] European HORIZON-MSCA-2021-SE-01-01 project CaLIGOLA - Cartan geometry, Lie and representation theory, Integrable Systems, quantum Groups and quantum computing towards the understanding of the geometry of deep Learning and its Applications; https://site.unibo.it/caligola/en

In physics, mathematics, economics, engineering, and many other fields, differential equations describe a function in terms of the derivatives of the variables. Put simply, when the rate of change of a variable in terms of other variables is involved, you will likely find a differential equation. Many examples describe these relationships. A differential equation's solution is typically derived through analytical or numerical methods.