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The paper is organized as follows. In Section we introduce the stochastic partial differential equations. We propose in Section the framework of stochastic collective coordinates, and apply it to the bistable equation with multiplicative noise in Section and with additive noise in Section . We present numerical results... |
2. Models We study here SPDEs for a single component [MATH] in one spatial dimension of the form [EQUATION] where the noise has zero mean [MATH] . Here the expectation is taken with respect to realizations of the driving noise process. As an explicit example, frequently considered in population dynamics, we consider a ... |
[EQUATION] with [MATH] , modelling the Allee effect when at low densities individual fitness increases with density . The resulting SPDE is known as the bistable equation or, in the context of action potential propagation on nerve fibres, as the Nagumo equation |
. The Nagumo equation supports in the noise-free case travelling waves with stable asymptotic states [MATH] and [MATH] . We consider additive and multiplicative noise. As multiplicative noise we consider noise which vanishes at the asymptotic states [MATH] and [MATH] . This prevents nucleation phenomena outside of the ... |
[EQUATION] with one-dimensional Brownian motion [MATH] . Note that the noise respects the underlying translational symmetry. As additive noise we consider white in time noise with spatial correlations prescribed by a covariance operator, [MATH] , with kernel |
[MATH] with finite trace. Then there exists a complete orthonormal basis [MATH] with [MATH] , and we can construct a [MATH] -Wiener process |
[EQUATION] with independent one-dimensional Brownian motions [MATH] satisfying [MATH] and [MATH] . We further assume that the noise is localized in space in a region [MATH] , thereby breaking the translational symmetry, which we model by |
[EQUATION] with [MATH] 3. Stochastic collective coordinates Initially developed for conservative deterministic nonlinear wave dynamics |
, the theory of collective coordinates has since been extended to dissipative deterministic systems such as reaction diffusion systems |
and recently to phase oscillators . Here we apply the method to the stochastic partial differential equations introduced in the previous section. |
Consider a parabolic semilinear SPDE of the form [EQUATION] with noise [MATH] with one-dimensional Brownian motion [MATH] and [MATH] [MATH] can be finite or infinite. We assume that the solution can be approximated by some ansatz function [MATH] for some time-dependent, so called collective coordinates [MATH] . The par... |
[EQUATION] where [MATH] is [MATH] -dimensional Brownian motion. Here the subscripts in the drift term [MATH] and in the [MATH] diffusion matrix [MATH] refer to the collective coordinates, i.e. [MATH] denotes the drift term for the collective coordinate [MATH] and [MATH] denotes the diffusive terms for the collective co... |
The ansatz function [MATH] does not satisfy the SPDE ( ) and inserting it into the SPDE defines a stochastic process via [EQUATION] |
which quantifies the error made by restricting the solution space to the ansatz function ( ) spanned by the collective coordinates. We remark that our definition of the error is different from the definition of the error as the (weighted) [MATH] norm of the difference [MATH] which is typically used in the literature on... |
[EQUATION] where we used Einstein’s summation convention. Substituting ( ) and collecting only terms up to order [MATH] we obtain, using the independence of the Brownian motion, |
[EQUATION] To maximize the degree to which the collective coordinates approximate solutions of the SPDE, we require that the stochastic process defined by [MATH] does not project onto the subspace spanned by the collective coordinates. We therefore require that the error [MATH] is orthogonal to the tangent space of the... |
[EQUATION] for [MATH] and the [MATH] diffusion contributions, which balance the Brownian motion of the SPDE with the Brownian motion of the collective coordinate system, are given by |
[EQUATION] for [MATH] . Here the angular brackets denote integration over the spatial domain as in [MATH] . For [MATH] and [MATH] for all [MATH] we can achieve pathwise matching provided |
[EQUATION] The [MATH] equations for the drift coefficients [MATH] 10 ) and the [MATH] diffusion coefficients ( 12 ) determine the drift and diffusion coefficients in the evolution equation for the collective coordinates ( ). Note that the noise in the system of SDEs for the collective coordinates is typically multiplic... |
In the case when [MATH] is infinite, one may want to consider finite [MATH] to reduce the complexity of the system of SDEs ( ) for the collective coordinates. Rather than achieving pathwise approximation of the solutions with [MATH] for all [MATH] , one may require instead that the statistical behaviour of the solution... |
[EQUATION] with [MATH] and [MATH] , the respective covariances of the two processes are also equal with [MATH] . The condition [MATH] then allows for a unique solution (up to the sign) of the diffusion matrix [MATH] of the collective coordinates as a matrix square root. Again, together with the [MATH] equations for the... |
If the SPDE is invariant under the action of some symmetry group, we expect deterministic behaviour of the collective coordinates associated with the shape dynamics for sufficiently small noise amplitudes; this is due to the shape dynamics being strongly contracting. On the other hand, the noise is unrestricted along t... |
We remark that the collective coordinate approach is different from the variational approach adopted in or those aimed at solving only for the wave speed |
or the interface location In the following we apply this general framework to the bistable stochastic partial differential equation introduced in the previous section. |
4. Collective coordinate approach for the bistable Nagumo equation with multiplicative noise Consider the bistable SPDE with multiplicative noise defined by ( |
[EQUATION] In order to find a dimension reduced description of the bistable Nagumo SPDE ( 13 ) we make the following ansatz [MATH] with |
[EQUATION] which defines the collective coordinates [MATH] . In the deterministic case this is an exact solution with [MATH] and [MATH] with [MATH] . We remark that it was shown in |
that such front solutions are stable in the bistable SPDE ( 13 ). The inverse front width [MATH] constitutes the shape dynamics whereas the location of the front interface [MATH] denotes the dynamics on the group orbit. Figure shows a snapshot of a travelling front for the bistable equation with multiplicative noise ( ... |
The collective coordinates encode the dynamics and are propagated via the system of SDEs [EQUATION] with one-dimensional Brownian motions [MATH] (cf. ( )). To determine the unknown drift and diffusion coefficients of the evolution equations ( 15 )–( 16 ) for the collective coordinates we require that the error made by ... |
[EQUATION] where we omit for ease of notation the hat to denote the ansatz function ( 14 ), and where [MATH] is given by ( ) for the bistable SPDE and [MATH] is the multiplicative noise defined in ( ). To maximize the degree to which the collective coordinates approximate the solution of the SPDE we set, as outlined in... |
[EQUATION] and [EQUATION] as well as diffusion contributions given by [EQUATION] and [EQUATION] The integrals can be explicitly calculated for the [MATH] -ansatz function ( 14 ) (see ). In particular the diffusion contributions ( 19 )–( 20 ) become |
[EQUATION] which is solved by [MATH] (i.e. no noise in the equation of the shape parameter [MATH] (cf. ( 15 )), and by setting [MATH] . This solution allows for pathwise approximation with [MATH] in the case when the stochastic forcing of the underlying SPDE ( 13 ) is known, or for the matching of the statistics of the... |
Evaluating the projections ( 17 )–( 18 ) determines the drift coefficients, and the evolution equation for the collective coordinates becomes the following skew-product system |
[EQUATION] with [MATH] and [MATH] being the stationary inverse width and constant front velocity of the deterministic bistable partial differential equation, and with Brownian motion [MATH] . The noise only enters the equation for the group variable and does not affect the shape variable [MATH] , consistent with the sy... |
[EQUATION] The inclusion of a fluctuating Allee threshold leads to a sharper interface [MATH] when compared to the deterministic travelling wave. This, maybe counter-intuitive, decrease in the interface width in the presence of noise can be explained by considering that fluctuations will push the front interface to the... |
[EQUATION] with superimposed fluctuations with constant variance [MATH] . The inclusion of noise hence leads to a slowing down of the front when compared to the deterministic speed [MATH] |
We remark that, upon expanding in [MATH] and letting [MATH] , we recover the result obtained in at order [MATH] with [MATH] and [MATH] . Our approach captures higher order effects and allows for the study of the temporal evolution. |
Furthermore, our results confirm previous detailed numerical results performed in where it was found that the wave becomes steeper with increasing noise and that the speed decreases linearly with [MATH] . This is consistent with our analytical results, which at [MATH] reduce to [MATH] and [MATH] |
4.1. Numerical results We now present numerical results demonstrating that the stochastic collective coordinate approach is capable of quantitatively describing the diffusive behaviour of travelling fronts. We present results for [MATH] [MATH] and [MATH] . We solve the SPDE ( 13 ) with multiplicative noise using an Eul... |
for the ansatz function ( 14 ). We have checked that linear interpolation yields similar results when used to determine the location of the interface [MATH] |
Figure shows a comparison of the phase of a travelling front of a simulation of the bistable SPDE ( 13 ) (defined as [MATH] such that the fitted [MATH] ) and its collective coordinate approximation provided by the system of SDEs ( 23 )–( 24 ) when we use the same stochastic forcing for both systems with [MATH] , demons... |
We now consider the case when the stochastic forcing of the SPDE is different from the stochastic forcing of the equations for the collective coordinates and [MATH] . We estimate the statistical behaviour of the inverse width of the travelling wave and its speed by computing temporal averages. We have checked that the ... |
Figure shows the empirical histogram of the inverse width of the travelling wave front of the bistable SPDE ( 13 ) with multiplicative noise. The mean [MATH] is approximated by the collective coordinate solution [MATH] (cf. ( 25 )), remarkably well with a relative error of [MATH] . The inverse width [MATH] experiences ... |
The average speed of the front is estimated as [MATH] and matches the result of the collective coordinates with its prediction [MATH] up to a relative error of [MATH] . The fluctuations around the linear drift can be estimated via the variance of [MATH] (after subtracting the mean linear drift). We find [MATH] is well ... |
For [MATH] the deterministic speed is [MATH] ; the collective coordinate ansatz predicts that the noise does not induce any non-zero speed corrections and has [MATH] (cf ( 26 )). This is confirmed in simulations of the full SPDE with an average speed of [MATH] ; the estimated value of the average speed decreases with t... |
5. Collective coordinate approach for the bistable Nagumo equation with additive noise Consider the bistable SPDE with additive noise defined in ( |
[EQUATION] driven by a [MATH] -Wiener process with kernel of the covariance operator [MATH] supported on a finite domain with [MATH] for [MATH] and [MATH] otherwise (which we approximate by the smooth function for [MATH] given by ( ) using [MATH] ). [MATH] and [MATH] represent the eigenvalues and eigenfunctions of the ... |
[EQUATION] For [MATH] the eigenvalue problem can be solved analytically on a finite domain of length [MATH] , and we have [EQUATION] |
with eigenvalues [EQUATION] where [MATH] and [MATH] are the solutions of the transcendental equations [EQUATION] The addition of spatially localized noise breaks the translational invariance. We employ again the [MATH] -ansatz function ( 14 ) as for the case of multiplicative noise and assume that the dynamics of the c... |
[EQUATION] This is only justified for small noise amplitude [MATH] and sufficiently small noise regions [MATH] , where the probability of noise induced nucleation of new fronts is small. We obtain again the projections ( 17 ) and ( 18 ), determining the drift coefficients [MATH] and [MATH] , which we can evaluate (usin... |
[EQUATION] The projections onto the stochastic terms yield, upon using [MATH] [EQUATION] Again pathwise matching can be achieved for [MATH] with diffusion coefficients |
[EQUATION] Note that when the front interface is well outside of the domain [MATH] , where [MATH] , the collective coordinates satisfy essentially deterministic evolution equations with [MATH] for all [MATH] . The breaking of the translational symmetry by the localized additive noise now allows for non-deterministic di... |
To avoid having to deal with infinitely many diffusion coefficients, we now present the calculations when truncating to [MATH] in ( 31 )–( 32 ), matching the mean and the variance of the now two-dimensional Brownian motion [MATH] of the collective coordinates with the statistics of the [MATH] -dimensional [MATH] -Wiene... |
[EQUATION] and [EQUATION] and, as outlined in Section , require that the implied diffusion coefficients of the two stochastic processes in ( 35 )–( 36 ) are equal with |
[EQUATION] The diffusion coefficients of the collective coordinates can then be determined as a square root, and give rise to multiplicative noise. From ( 38 ) we can immediately derive |
[EQUATION] which affect the drift coefficients ( 33 )–( 34 ) and also determine the variance of increments of the collective coordinates with [MATH] and [MATH] (upon subtracting the linear drift from [MATH] ). |
Since the diffusion coefficients depend on [MATH] and [MATH] we are not able to find an explicit solution for the expected value of the inverse width [MATH] by solving for [MATH] in ( 33 ). However, for [MATH] we can approximate [MATH] . Similarly we find that [MATH] (cf ( 34 )) in the small noise limit. Recall that we... |
5.1. Numerical results We now present the numerical results for the case of additive noise and compare results from simulating the Nagumo equation ( 27 ) with additive noise with the analytical results from the collective coordinate approach where we employ here the expressions for the diffusion coefficients ( 37 ) yie... |
. This method produces complex valued noise, of which the real and imaginary part are independent and identically distributed (only the real part is used in our simulations). The infinite-sum of the [MATH] -Wiener process in ( 27 ) is truncated to [MATH] where [MATH] . An outline of the method is included in . The SPDE... |
Figure shows the phase [MATH] and the inverse width [MATH] of a travelling front obtained from a simulation of the bistable SPDE ( 27 ) with additive noise as well the respective collective coordinate approximations provided by the system of SDEs ( 31 )–( 32 ) with the drift coefficients given by ( 33 )–( 34 ) and diff... |
We consider now the case when the stochastic forcing of the SPDE is different from the stochastic forcing of the equations for the collective coordinates and [MATH] . The mean inverse width of the full SPDE is estimated as [MATH] , and is well approximated by the deterministic limit of the collective coordinates approa... |
[MATH] for the SPDE and with [MATH] for the collective coordinates, with a relative error of [MATH] . Similarly, the variance [MATH] for the SPDE is reproduced well by the collective coordinate approach which predicts [MATH] , implying a relative error of [MATH] The fluctuations around the mean inverse width [MATH] and... |
6. Discussion and outlook Employing a symmetry perspective whereby the dynamics of a travelling front is split into the shape dynamics and the dynamics along the symmetry group, we introduced a collective coordinate framework to reduce an infinite dimensional SPDE to a system of finite dimensional SDEs. This allowed us... |
It is pertinent to mention that the actual form of the ansatz function ( 14 ) is not relevant. We have also performed the calculation for an ansatz function [MATH] and an ansatz function [MATH] ; the accuracy was tested against simulations of the SPDE exhibiting the same order of magnitude in relative errors. The resul... |
The method introduced here is not limited to travelling fronts. By choosing bell-shaped test functions rather than the front solutions explored in this work, one may study the front selection problem and propagation failure in stochastic systems supporting pulse solutions using collective coordinates as was done in the... |
The symmetry perspective is not restricted to models with translational symmetry but can also be applied to pattern forming systems on the plane exhibiting Euclidean symmetry involving translations and rotations supporting spiral waves. This is relevant to stochastically perturbed excitable media. Spiral waves in excit... |
The symmetry perspective has been successfully used in a numerical algorithm to simulate travelling waves in SPDEs . Their algorithm, however, does not involve explicit expressions for the dynamics on the group, but the dynamics is determined numerically. To explicitly incorporate the dynamics of the collective coordin... |
might lead to a more efficient numerical method to simulate SPDEs with symmetry. Our work poses important theoretical questions. Whereas the symmetry perspective is well studied in the case of deterministic PDEs and conditions on its validity are well established (see for example |
and references therein), the extension into the stochastic realm of SPDEs with symmetry is only now developing with first promising rigorous results on front propagation with translational symmetry where a phase equation is sought to control the neutral direction |
. In the case of multiplicative noise for the Nagumo equation this leads to the same equations for the equilibrium shape and the phase we found in ( 25 )–( 26 ). |
Acknowledgments We thank Ben Goldys, Gabriel Lord, James Maclaurin and Gilles Vilmart for stimulating discussions. GAG acknowledges funding from the Australian Research Council, grant DP180101991. |
Appendix A Explicit formulae for the collective coordinate projections We list here several integrals which appear in the evaluations of the projection when using the [MATH] -ansatz function |
[EQUATION] Using [MATH] and [MATH] , we evaluate [EQUATION] The following integrals, which appear in the projections, can be analytically determined |
[EQUATION] Appendix B Construction of [MATH] -Wiener noise Following closely , we briefly describe how to generate a [MATH] -Wiener process, denoted in this section by [MATH] , with kernel of the covariance operator [MATH] with finite trace. That is we need to find a complete orthonormal basis [MATH] as eigenfunctions ... |
[EQUATION] with independent one-dimensional Brownian motions [MATH] Denote by [MATH] the [MATH] covariance matrix with [MATH] with [MATH] , approximating the covariance operator [MATH] and denote by [MATH] the numerical approximation of the increments [MATH] of the [MATH] -Wiener process. We seek [MATH] (in the main te... |
[EQUATION] where [MATH] . Suppose further that the spectral decomposition of [MATH] is [MATH] , where [MATH] is the orthonormal matrix with columns [MATH] being the eigenvectors of [MATH] , and [MATH] is the diagonal matrix with eigenvalues [MATH] . Let [MATH] . Then |
[EQUATION] A straightforward method to generate [MATH] would be to find the eigenvectors and associated eigenvalues of the covariance matrix [MATH] , and truncate to [MATH] eigenvectors and corresponding eigenvalues. However, this has complexity of [MATH] to compute the eigen-decomposition. Computationally even more ex... |
). We note that as the correlation length [MATH] decreases, the eigenvalues of [MATH] decay more slowly so large values of [MATH] are required to reliably approximate the [MATH] -Wiener process. |
We now describe a more efficient way to reduce the numerical complexity to [MATH] operations using the fast Fourier transform (FFT) by embedding the covariance matrix [MATH] into a higher-dimensional circulant [MATH] matrix. Let us define the following classes of matrices: |
Definition (Toeplitz matrix) A Toeplitz matrix is an [MATH] real valued matrix [MATH] where [MATH] for some real numbers [MATH] Definition |
(Circulant matrix) A Toeplitz matrix [MATH] is circulant if [MATH] for [MATH] and [MATH] for [MATH] Symmetric Toeplitz matrices have [MATH] , and symmetric circulant matrices have [MATH] . Circulant matrices [MATH] are of the following general form |
[EQUATION] One can diagonalize a real-valued circulant [MATH] matrix [MATH] with first column [MATH] as [MATH] , where [MATH] is the complex Fourier matrix (equivalent to the FFT) with entries [MATH] where [MATH] , and [MATH] is a diagonal with diagonal elements [MATH] . For details the interested reader is referred to |
In our application the covariance matrix for the [MATH] -Wiener noise is [MATH] and we have [MATH] and [MATH] . Therefore the covariance matrix [MATH] is symmetric and Toeplitz, but it is not circulant. By embedding [MATH] inside a larger circulant matrix [MATH] , with |
[EQUATION] where [MATH] and [MATH] are such that [MATH] is circulant (i.e. of the form ( 42 )). The larger circulant matrix [MATH] has the decomposition [MATH] [MATH] -Wiener increments [MATH] with mean 0 can be [MATH] , where [MATH] is a complex normal random variable, with [MATH] . Note that [MATH] has independent id... |
Finally, we truncate [MATH] to its first [MATH] entries, and define [MATH] . The covariance matrix of [MATH] is therefore [MATH] , and we have the required [MATH] -Wiener increment. |
# Source: arxiv 1806.07268 # Title: Beyond Local Nash Equilibria for Adversarial Networks # Sections: all # Downloaded: 2026-03-02T09:23:17.128723+00:00 |
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Beyond Local Nash Equilibria for Adversarial Networks Abstract Save for some special cases, current training methods for Generative Adversarial Networks (GANs) are at best guaranteed to converge to a ‘local Nash equilibrium’ (LNE). Such LNEs, however, can be arbitrarily far from an actual Nash equilibrium (NE), which i... |
resource-bounded Nash equilibrium (RB-NE) : by increasing computational resources we can find better solutions. We empirically demonstrate that our method is less prone to typical GAN problems such as mode collapse, and produces solutions that are less exploitable than those produced by GANs and MGANs, and closely rese... |
Introduction Generative Adversarial Networks (GANs) (Goodfellow et al., 2014 are a framework in which two neural networks compete with each other: the generator (G) tries to trick the classifier (C) |
into classifying its generated fake data as true. GANs hold great promise for the development of accurate generative models for complex distributions. Consequently, in just a few years, GANs have grown into a major topic of research in machine learning. A core appeal is that they do not need to rely on distance metrics... |
(Unterthiner et al., 2018 ; Arjovsky and Bottou, 2017 ; Arjovsky et al., 2017 . A typical problem is mode collapse , which can take the form of |
mode omission , where the generator does not produce points from certain modes, or mode degeneration , in which for at least one mode the generator only partially covers the mode. Moreover, while learning the players may forget : e.g., it is possible that a classifier correctly learns to classify part of the input spac... |
(Heusel et al., 2017 ; Unterthiner et al., 2018 is a guarantee to converge to a local Nash equilibrium (LNE) (Ratliff et al., 2013 . However, an LNE can be arbitrarily far from an NE (which we will also refer to as ‘global NE’ to discriminate) and the corresponding generator might be exploitable by a strong opponent du... |
We introduce a novel approach that does not suffer from getting trapped in LNEs: finite Generative Adversarial Network Games (GANGs) formulate adversarial networks as finite zero-sum games, and the solutions that we try to find are saddle points in mixed strategies . This approach is motivated by the observation that, ... |
Since we cannot expect to find exact best responses due to the extremely large number of pure strategies that result for sensible choices of neural network classes, we introduce resource-bounded best-responses (RBBRs), and the corresponding resource-bounded Nash equilibrium (RB-NE), which is a pair of mixed strategies ... |
The key features of our approach are that: It is based on finite zero-sum games, and as such it enables the use of existing game-theoretic methods. In this paper we focus on one such method, Parallel Nash Memory (PNM) (Oliehoek et al., 2006 |
It will not get trapped in LNEs: we prove that it monotonically converges to an RB-NE, which means that more computation can improve solution quality. |
Moreover, it works for any network architecture (unlike previous approaches, see Section ). In particular, future improvements in classifiers/generator networks can be exploited directly. |
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