We address a computational framework that integrates symmetry reduction into Physics-Informed Neural Networks (PINNs) for analysing symmetry-driven dynamics in nonlinear partial differential equations (PDEs). Using an auxiliary network to learn time-dependent transformations, we render the symmetry-invariant solutions stationary or slowly varying in rescaled coordinates while simultaneously inferring the symmetry parameters (e.g., wave speed, scaling rates). This yields a modified evolution equation coupled with algebraic constraints on symmetry parameters, producing index-2 differential-algebraic equation (DAE) systems. Since conventional standard PDE/ODE solvers struggle or even fail with such high-index DAEs, we employ PINNs as an alternative approach that naturally unifies PDE residuals and algebraic constraints in a single loss function. This allows simultaneous inference of the invariant solutions and the transformation properties without large domains, mesh adaptivity, or front tracking. Beyond forward simulation, our framework further enables robust parameter inference from sparse, spatially offset data where vanilla PINNs fail. Our numerical demonstrations include, among others, the 2D porous medium, the generalised Korteweg-de Vries and Burgers PDE, showcasing our proposed approach as a powerful tool for the solution of both the forward and inverse problems for index-2 DAEs arising in nonlinear wave and scaling dynamics. Many nonlinear partial differential equations exhibit dynamics governed by continuous symmetries. Here, authors integrate symmetry reduction into physics-informed neural networks, enabling accurate forward simulation and parameter inference without front tracking or mesh adaptation.
M. Kavousanakis, Gianluca Fabiani, A. Georgiou et al.· Nature Communications· 0 citations
We study the formation and interaction of dispersive shock waves (DSWs) in the two-dimensional Toda lattice subject to wedge-type initial conditions, and show that their interaction gives rise to a discrete analog of Mach reflection for dispersive shock waves in discrete systems. The initial jump across each leg of the wedge acts locally as a Riemann problem for the one-dimensional Toda lattice, producing two oblique DSWs whose leading-edge soliton amplitude is determined explicitly by the one-dimensional Whitham modulation theory. The two-dimensional nature of the problem manifests when these oblique DSWs meet along the symmetry axis. We show that, for compressive wedges (i.e., when the initial conditions are such that two oblique DSWs that are generated propagate toward each other), a critical slope $q_{\mathrm{cr}}$ separates two regimes: in the subcritical regime ($q<q_{\mathrm{cr}}$) the interaction is resonant and it produces an expanding DSW whose amplitude, length and velocity are explicitly computed by using exact soliton solutions of the two-dimensional Toda lattice; in the supercritical regime ($q>q_{\mathrm{cr}}$) the interaction is ordinary and produces a localized peak whose amplitude is determined analytically. We also show qualitatively that a similar dichotomy between two regimes exists for expansive wedges (i.e., when the initial conditions are such that the two oblique DSWs propagate away from each other). We confirm all analytical predictions by comparing them with the results of direct numerical simulations. Finally, we show that the continuum limit of the result is consistent with the analogous theory for the Kadomtsev-Petviashvili equation, providing an independent validation of the analytical framework.
M. Calabrese, G. Biondini, C. Chong et al.· 0 citations
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