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[EQUATION] The framework may be extended easily to arbitrary observation functions and likelihoods (including Gaussian Processes models of the observation function), as long as we can compute the probability [MATH] , estimate its gradient with respect to [MATH] and infer an approximation to the posterior [MATH] |
3.1 Sparse Gaussian process transition function inference and learning We represent the scalar-valued stochastic transition functions [MATH] as independent Gaussian Processes (GP), such that the latent map is [MATH] , and we can easily query the predicted mean and variance of the map for any given input. Unforunately, ... |
The process [MATH] is thus a Sparse Gaussian Process (SGP) parametrised by: a positive definite kernel function [MATH] ; the inducing point locations [MATH] ; and the uncertain values at those locations, represented as random variables drawn from [MATH] . The ability to represent different noise levels at different loc... |
[EQUATION] We then recover the vector-valued transition function as a collection of the scalar functions, [MATH] . The learning of the transition function thus boils down to the estimation of the inducing point parameters [MATH] . Often the kernel function itself is parametrised, [MATH] , in which case we may choose to... |
3.1.1 Inference In order to carry out the inference of a latent trajectory, given the data and the current estimate of the transition and observation functions, there are various algorithms available (Wan & Merwe, 2000 . We choose to use Assumed Density Filtering (Ramakrishnan et al., 2011 here, based on empirical perf... |
We represent our belief of the latent state at time [MATH] as a normal distribution with mean [MATH] and diagonal covariance matrix [MATH] . We first need to propagate our belief through the transition function, to get an estimate of our updated belief, which we approximate as a normal distribution [MATH] , with a non-... |
[EQUATION] where the required expectations may be computed in closed form for linear and Exponentiated Quadratic kernels, and are shown in Appendix B. |
Given our belief of the latent state at time [MATH] , we need to incorporate the data into our belief to obtain the moments. Thanks to our simple linear-Gaussian observation model, and our approximate belief, this can be done exactly: |
[EQUATION] We then approximate the covariance matrix [MATH] with its diagonal, as per moment matching, and proceed to carry out the filtering for the next time step, until the complete trajectory has been recovered. The parameters of the latent model we require for inference are [MATH] |
3.1.2 Learning In order to obtain a good estimate of the parameters of our model, we need to be able to learn them. There are many frameworks available to carry out this estimation (Titsias, 2009 ; Titsias & Lawrence, 2010 ; Bui et al., 2016 , we chose - again, based on empirical evidence - to use gradient ascent with ... |
[EQUATION] Equipped with this objective function, and the fact, that our model is differentiable with respect to all of its parameters, we may optimise our parameters via gradient ascent. We may then iterate inference and learning steps until convergence. |
3.2 Conditioning on fixed points We now wish to extend our framework towards a Fixed Point Sparse Gaussian Process, which provides explicit representation of the learned map’s fixed points and their local linearisation as parameters of the model. |
We may think of a fixed point in a Sparse Gaussian Process as a special inducing point, whose value is tied to its location. Furthermore, to represent and inquire about the stability of said fixed point, we wish to attach derivative observations to that location, representing the local Jacobian. This way we may uncover... |
The steps we need to go through for the derivation of the system largely follow what has been described in the previous section, with a few extra complications. Let the fixed points be represented as random variables drawn from [MATH] , where the [MATH] variables may be used by the system to disable unnecessary fixed p... |
We can thus extend the parameter set describing our current belief of the map, [MATH] . We use this set of parameters to carry out the inference, requiring us to revisit eqs. |
to 10 . The main change comes from the fact, that we wish to use the derivative observations attached to our fixed points during inference. This requires establishing a derivative Gaussian Process, whose kernel function is given by the derivative of the original kernel function with respect to both arguments |
[EQUATION] resulting in a fourth order tensor. It is useful to define a block-structure matrix version [MATH] . We may similarly define the cross-covariance between a normal and a derivative process, as taking the derivative with only to the respective argument of the original kernel function: |
[EQUATION] Equipped with these processes, we may re-write the predictive eqs. to 10 as the block matrices [EQUATION] [EQUATION] [EQUATION] |
Resulting in the predictive moments for a noiseless input: [EQUATION] where [MATH] and [MATH] represent the concatenation of indices [MATH] . For the inference we still need to consider propagating beliefs represented as Gaussian random variables, thus requiring to compute [MATH] and [MATH] . For the linear and the Exp... |
The learning does not change significantly, our objective function remains the same, and our operations remain differentiable with respect to all parameters. |
Experiments Equipped with fully described model, we are ready to test it. As our first example we are going to return to the well-studied example described in section |
. Finally, we study changes of fixed point pattens in an influential model of mutually inhibiting neural populations during decision making (Machens et al., 2005 |
4.1 Stochastic pitchfork bifurcation We may now write down the stochastic version of eq. [EQUATION] where [MATH] iid. We examine, how varying [MATH] affects the learned fixed points. We trained the system using 32 trials, lasting 20 time steps each, with [MATH] , and the initial condition [MATH] . Note that this is les... |
C. We then fit our model to the data with 16 inducing points and the overestimated 5 fixed points, letting the ARD formulation determine the number of fixed points present in the system. |
We first confirmed, that the method indeed captures the available data very well for various values of the bifurcation parameter [MATH] , as shown in, Figure |
A and B. We then create the bifurcation plot, Figure C, based on the learned parameter values. The fixed points identified truthfully track the expected location and stability, as well as successfully recovering the true number of fixed points. Consistently with previous finding on similar systems (Diks, 2006 , we inde... |
4.2 Mutually inhibiting neural populations Having recovered previous results with our highly flexible system, we now turn our attention to a system closer to the data-analysis-in-the-wild type problems we aimed to solve. |
The data used comes from a simulated system, but one that was optimised to match the behaviour of measured neural population. For more details about the experiment and the simulation, read the excellent paper from Machens, Romo and Brody 2005 |
In short the system consists of an external excitation and so-called negative and positive populations. In the current study we do not take into account possible differential inputs to the populations, so for our purposes the model is completely symmetric. We slightly reformulate the system equations to match the langu... |
[EQUATION] where [MATH] is iid noise, [MATH] is the inhibitory weight of the populations, and importantly, [MATH] is a numerically optimised function, which defines the nullclines of the system and gives rise to interesting system behaviour, as the external driving input [MATH] - our bifurcation parameter - is varied. |
The nullclines generally cross one another, giving rise to fixed points, whose stability depend on the angle of crossing. In particular, the original study finds that in one extreme, two stable fixed points are created far from one another, and may be used for decision making, whereas during the other, there is only a ... |
Note that this model was carefully designed to behave so, and our goal here is to estimate such behaviour purely from data [MATH] |
Examining the results in Figure A-C the system behaviour was very well captured by the estimated fixed points, including the number, location and stability of the points. Although the stability of the system is inferred correctly, if we examine the eigenspectrum of the central point in Figure |
D, we can indeed follow its stabilisation from a saddle with one stable and one unstable direction to a mostly stable fixed point. |
Discussion Studying real systems, especially in biological experiments, where we have little knowledge of the governing equation is a hard but ubiquitous problem. In the current study we designed an algorithm aimed at the study of random dynamical systems measured at discrete time, in which we can modify or measure a v... |
All analyses comes with limitations of course. The current work is aimed at fixed point bifurcations, at the moment we can not sufficiently describe limit cycles or more than zero-dimensional attractors, beyond a rudimentary approximation in the form of aligned fixed points. Another issue that often comes up in learnin... |
Our core contribution is the Fixed Point Sparse Gaussian Process formulation, in which fixed points appear explicitly as parameters of the model fit to data, and may thus be identified directly by parameter optimisation methods. This core idea has many potential extensions highlighted throughout the paper, including ex... |
Furthermore, combining this powerful stochastic representation with the ability to robustly identify fixed points in unknown systems may indeed bring further effort into stochastic bifurcation analysis, an exciting and very powerful methodology, still in its infancy. |
# Source: arxiv 1807.01610 # Title: On the geometry of the Clairin theory of conditional symmetries for higher-order systems of PDEs with applications # Sections: all # Downloaded: 2026-03-02T08:53:47.446531+00:00 |
On the geometry of the Clairin theory of conditional symmetries for higher-order systems of PDEs with applications Abstract. This work presents a geometrical formulation of the Clairin theory of conditional symmetries for higher-order systems of partial differential equations (PDEs). We devise methods for obtaining Lie... |
Key words and phrases: Lie point symmetries, conditional symmetries, contact forms, PDE Lie systems, jet bundles, Clairin formalism, nonlinear wave equation, Gauss–Codazzi equations, generalised Liouville equation |
1991 Mathematics Subject Classification: 35Q53 (primary); 35Q58, 53A05 (secondary) 1. Introduction Over the last two centuries, Lie’s theory of symmetries of partial differential equations (PDEs) |
has been the subject of extensive research in mathematics and physics . During those years, the development of this theory has led to significant progress in classifying and solving differential equations, yielding many new interesting results (see e.g. |
and references therein). A number of attempts to generalise this subject and to develop its applications can be found in the literature |
Of particular interest from a physical point of view has been the development of the theory of conditional symmetries which evolved in the process of extending Lie’s classical theory of symmetries of PDEs |
. This approach consists essentially of supplementing the original system of PDEs with certain first-order differential constraints (DCs) for which a symmetry criterion is applied. As a result we obtain an overdetermined system of PDEs admitting, in some cases, a larger class of Lie point symmetries than the original s... |
in 1903 by subjecting the original system of PDEs to several DCs. Since then, generalisations of conditional symmetries have been formulated by many authors |
In the Clairin theory of conditional symmetries the DCs are given by an integrable (in the sense of fulfilling a compatibility condition explained later and in |
) first-order system of PDEs in normal form , i.e. such that the derivatives of particular solutions are functions of the dependent and independent variables, and compatible with the initial system of PDEs. Although there exist more general conditional symmetry methods based on adding DCs that need not give rise to fir... |
, the Clairin theory allows us to study the solutions of the DCs through many techniques. For instance, one can use different types of Lie systems |
or a naturally related Abelian Lie algebra of Lie point symmetries, which generates new particular solutions of the initial system of PDEs from known ones satisfying the given DCs. |
In the Clairin theory, the DCs are determined by the zeroes of the characteristics of a Lie algebra [MATH] of vector fields . The Lie algebra [MATH] consists of Lie symmetries, referred to as Clairin conditional symmetries , of the overdetermined system formed by the original system subjected to such DCs. The crux of t... |
. To simplify the task, [MATH] is assumed to be Abelian and to admit a basis of a particular type . Then, [MATH] can be obtained by solving a nonlinear system of PDEs |
The Clairin method, which is mainly described in terms of coordinates (cf. ), lacks an intrinsic geometric formulation. Additionally, there exists to our knowledge no detailed geometric formalism for studying higher-order systems of PDEs in this context, as works have, until now, focused on the formalism for first-orde... |
or on applying conditional symmetry techniques to higher-order systems of PDEs without a detailed theoretical analysis Hence, our first aim is to present a geometric Clairin theory of conditional symmetries for higher-order systems of PDEs. This allows us to avoid most previous assumptions on [MATH] , to clarify some o... |
, and, as a consequence, to provide new insights into other conditional symmetries and related structures . Our theory also provides methods for constructing new Lie algebras of conditional symmetries from known ones. Our second aim is to develop methods for obtaining conditional symmetries through the so-called PDE Li... |
Geometrically, the Clairin theory of conditional symmetries for higher-order systems of PDEs can be summarised as follows. An [MATH] -th order system of PDEs whose dependent and independent variables are functions on [MATH] and [MATH] , respectively, amounts to a subset [MATH] of the [MATH] -th order jet bundle, [MATH]... |
. Lie algebras of conditional symmetries of [MATH] are given by a Lie algebra [MATH] of vector fields on [MATH] defining a submanifold [MATH] , the so-called characteristic system of [MATH] , given by the common zeroes of the total differentials |
of the characteristics of the elements of [MATH] in such a way that [MATH] consists of Lie point symmetries of the system of PDEs related to [MATH] . The Clairin theory requires that [MATH] be related to a first-order system of PDEs in normal form satisfying integrability conditions (sometimes called the compatibility ... |
. If [MATH] is Abelian and admits a basis of a particular type , then [MATH] can be derived by solving a nonlinear system of PDEs |
Apart from providing a careful geometric Clairin theory for conditional symmetries of higher-order systems of PDEs, let us describe other new contributions of our work. |
First, by describing the characteristics of vector fields via the contact forms on [MATH] , the characteristic system [MATH] for a linear space [MATH] of vector fields on [MATH] is given in Definition 3.1 or, equivalently, in Definition 3.2 in an intrinsic geometrical way. This geometrises and generalises several types... |
Theorems 4.3 and 4.5 characterise when [MATH] can be described as a system of PDEs in normal form. If [MATH] is assumed to be a system of PDEs in normal form, then Theorem 5.3 provides necessary and sufficient conditions to ensure that the prolongations to [MATH] (see |
) of the vector fields of [MATH] are tangent to [MATH] . This fulfils results given in the previous literature, where only necessary conditions are detailed |
. Remarkably, [MATH] need not be a Lie algebra, as assumed previously . Since our results state that [MATH] must span a distribution [MATH] of dimension [MATH] projecting onto [MATH] , one obtains that this condition, in coordinates, amounts to the existence of a non-degenerate [MATH] matrix [MATH] appearing in the Cla... |
Subsequently, Proposition 5.5 details the necessary and sufficient conditions on [MATH] to ensure that [MATH] , related to the system in normal form, is locally solvable |
. Corollary 6.6 shows that the standard conditions found in the literature ensure that [MATH] is locally solvable. Remarkably, if [MATH] , then Theorem 4.5 and Proposition 5.5 ensure that if [MATH] is a system of PDEs in normal form, then [MATH] is locally solvable, which also implies that the projection of [MATH] to [... |
. This implies that the solutions of [MATH] are the same as those of the system of PDEs given by [MATH] (where [MATH] is considered as a subspace of [MATH] in the natural way |
). This is the usual approach appearing in the literature (see and Section ). The Clairin approach to conditional symmetries assumes that [MATH] has a basis of a particular form |
. Section characterises geometrically when [MATH] admits such a basis. Such Lie algebras are called rectified PDE Lie algebras . Next, it is studied when a linear space of vector fields [MATH] is such that [MATH] . If [MATH] is a Lie algebra, then it is called a rectifiable PDE Lie algebra |
Next, we survey the geometric properties of conditional Lie symmetries and raise certain technical questions frequently overlooked in the literature. We show that every normal PDE Lie algebra of conditional symmetries [MATH] is such that every rectifiable PDE Lie algebra of the vector fields [MATH] satisfying [MATH] is... |
Finally, we study the differential equations characterising rectified PDE Lie algebras of conditional symmetries. To solve them, we extend the methods of |
and provide conditions to ensure that rectified PDE Lie algebras can be derived via the so-called PDE Lie systems . PDE Lie systems are first-order systems of PDEs in normal form admitting a superposition rule , i.e. a function allowing us to obtain their general solutions in terms of a generic family of particular sol... |
. These systems of PDEs have attracted much attention lately and their special structure allows for the development of methods to study their solutions |
Our use of PDE Lie systems to obtain Lie algebras of conditional symmetries is more general than the approach used in , where only particular types of PDE Lie systems or standard Lie systems appear. Moreover, we provide assumptions on higher-order systems of PDEs, [MATH] , that enable us to construct a PDE Lie system d... |
As applications, PDE Lie systems are employed to study nonlinear wave equations and minimal surfaces for Gauss–Codazzi equations |
. The fact that PDE Lie systems related to solvable Vessiot–Guldberg Lie algebras can be solved is employed to obtain minimal surfaces of Gauss–Codazzi equations (cf. |
). Finally, some solutions of the generalised Liouville equations are provided. This paper is organised as follows. Section 2 describes the basic geometric tools employed in the work. Section 3 is concerned with the definition of a characteristic system for a linear space of vector fields on [MATH] . Section 4 establis... |
2. Geometric preliminaries on jet bundles and systems of PDEs This section presents the notions of the theory of jet bundles and the notation to be employed in this work. This allows us to make our work more self-contained and easier to follow. Unless otherwise stated, we assume that mathematical structures are smooth ... |
Let [MATH] and [MATH] be manifolds of dimension [MATH] and [MATH] , respectively. We define [MATH] and [MATH] , where [MATH] stands for the [MATH] -th order jet bundle of the trivial bundle given by the projection map [MATH] onto [MATH] . Let [MATH] and [MATH] be coordinate systems on [MATH] and [MATH] , respectively. ... |
If [MATH] are integers and [MATH] , then [MATH] is a bundle over [MATH] relative to the natural projection [MATH] and we write [MATH] for its space of sections. |
We denote by [MATH] an arbitrary point of [MATH] . Given a section [MATH] , say [MATH] , its prolongation to [MATH] is the section [MATH] of the form [MATH] for [MATH] . Sections [MATH] of the form [MATH] for a [MATH] are called holonomic |
The main geometric structure on [MATH] , the so-called Cartan distribution [MATH] , is the smallest distribution on [MATH] tangent to all prolongations [MATH] for an arbitrary [MATH] |
. In coordinates, [MATH] is spanned by the vector fields [EQUATION] where [MATH] [MATH] [MATH] , and [MATH] is called the total derivative relative to [MATH] on [MATH] . The Cartan distributions are not involutive |
We write [MATH] for a multi-index [MATH] and [MATH] when [MATH] contact form on [MATH] is a one-form [MATH] on [MATH] such that [MATH] for every [MATH] . Hence, contact forms allow us to determine when a section [MATH] is holonomic. We write [MATH] for the space of contact forms on [MATH] . In particular, the one-forms |
[EQUATION] are called the basic contact forms on [MATH] relative to the coordinate system [MATH] , with [MATH] . On an open set of [MATH] with coordinates [MATH] , contact forms are linear combinations with functions on [MATH] of ( 2.1 ). In this sense, the basic contact forms ( 2.1 ) form a basis of [MATH] |
A vector field [MATH] on [MATH] can be written in local coordinates as [EQUATION] where [MATH] are univocally defined functions on [MATH] . The prolongation of [MATH] to [MATH] is the only vector field, [MATH] , on [MATH] leaving the space of vector fields taking values in [MATH] invariant (relative to the Lie bracket ... |
[EQUATION] where the functions [MATH] are the so-called characteristics of the vector field [MATH] . Geometrically, [EQUATION] where [MATH] stands for the contraction of a one-form [MATH] with a vector field [MATH] |
First-order systems of PDEs in [MATH] independent and [MATH] dependent variables are defined by [EQUATION] where [MATH] for [MATH] are certain functions. A particular solution of ( 2.4 ) is a map [MATH] from [MATH] to [MATH] whose associated section [MATH] is such that its prolongation to [MATH] satisfies ( 2.4 ). |
The system of PDEs ( 2.4 ) determines a region [MATH] where all the functions [MATH] , with [MATH] , vanish simultaneously. As is standard in the literature |
, it is hereafter assumed that system ( 2.4 ) has maximal rank , i.e. the functions [MATH] are functionally independent and [MATH] can be considered as a submanifold of [MATH] (cf. 33 , p. 158] ). A system of PDEs ( 2.4 ) is locally solvable if for each [MATH] there exists a solution [MATH] of the system ( 2.4 ) such t... |
33 , p. 158] . Finally, we also assume that [MATH] , which ensures that there exists at least one solution of [MATH] defined around every [MATH] |
A large family of differential equations is locally solvable and has maximal rank. To understand the generality of the local solvability assumption, we provide the following proposition characterising local solvability of first-order systems of PDEs in normal form. Similar results, with fewer details, can be found in |
Proposition 2.1 A first-order system of PDEs in normal form [EQUATION] is locally solvable if and only if it is integrable, i.e. [MATH] for every [MATH] and [MATH] |
. In turn, its integrability amounts to the fact that the Lie algebra of vector fields spanned by [EQUATION] is Abelian. Proof. Geometrically, the system ( 2.5 ) amounts to a submanifold [MATH] given by points of the form [MATH] of [MATH] . The system ( 2.5 ) has a particular solution passing through each point [MATH] ... |
[EQUATION] The latter amounts to the condition [MATH] for every [MATH] and [MATH] . Hence, system ( 2.5 ) is integrable. Note that the integrability of system ( 2.5 ), namely ( 2.7 ), amounts to |
[EQUATION] Hence, if ( 2.5 ) is integrable, then the distribution spanned by [MATH] is integrable and its integral submanifolds give solutions of ( 2.5 ) for every point [MATH] . This implies that ( 2.5 ) is locally solvable. |
A vector field [MATH] on [MATH] is a Lie point symmetry of the system of PDEs ( 2.4 ) if [EQUATION] As the prolongation to [MATH] of the Lie bracket of two vector fields on [MATH] is the Lie bracket of their prolongations to [MATH] |
, the Lie bracket of two Lie point symmetries of [MATH] is a new Lie point symmetry. Thus, Lie point symmetries form a Lie algebra [MATH] , which, when finite-dimensional, locally defines a Lie group action on [MATH] . This Lie group action transforms solutions of ( 2.4 ) into solutions of the same equation and allows ... |
3. A definition of characteristic systems for linear spaces of vector fields The description of Lie algebras of conditional symmetries for systems of PDEs is based on the geometry of the characteristic system , namely a submanifold of [MATH] to which we restrict the study of solutions of our initial system of PDEs |
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