In this Appendix, we will derive the fixed-point equations for the order parameters presented in the main text, following and generalising the analysis in Ref. [ Saddle-point equations The saddle-point equations are derived straightforwardly from the obtained free energy functionally extremising with respect to all parameters. The zero-regularisation limit of the logistic loss can help us study the separability transition. N 5 + \ 1 p 0, 1 d 5. (66) As a result, given that \ 2( 0, 1 ], the smaller value for which E is finite is U This result has been generalised immediately afterwards by Pesce et al. Ref. [ 59 ] for the Gaussian case, we can obtain the following fixed-point equations, 8 > > > > > >< > > > > > >: E = Mean universality Following Ref. [ In our case, this condition is simpler than in Ref. [ We see that mean-independence in this setting is indeed verified. Numerical experiments Numerical experiments regarding the quadratic loss with ridge regularisation were performed by computing the Moore-Penrose pseudoinverse solution.

Spiking neural networks (SNNs) are brain-inspired models that enable energy-efficient implementation on neuromorphic hardware. However, the supervised training of SNNs remains a hard problem due to the discontinuity of the spiking neuron model.

They generalize classic feedforward models and are equivalent toinfinite-depth weight-tied feedforwardnetworks.

We present a discretization-free scalable framework for solving a large class of mass-conserving partial differential equations (PDEs), including the time-dependent Fokker-Planck equation and the Wasserstein gradient flow. The main observation is that the time-varying velocity field of the PDE solution needs to be self-consistent: it must satisfy a fixed-point equation involving the probability flow characterized by the same velocity field. Instead of directly minimizing the residual of the fixed-point equation with neural parameterization, we use an iterative formulation with a biased gradient estimator that bypasses significant computational obstacles with strong empirical performance. Compared to existing approaches, our method does not suffer from temporal or spatial discretization, covers a wider range of PDEs, and scales to high dimensions. Experimentally, our method recovers analytical solutions accurately when they are available and achieves superior performance in high dimensions with less training time compared to alternatives.

Input: Network parameters θ; Input data x; Label y; Time steps T; Other hyperparameters; Output: Trained network parameters θ . Calculate the output o and the loss L based on o and y . Update θ based on the gradient-based optimizer. We first prove Theorem 1. Then Theorem 2 is similarly proved. We omit repetitive details here.

Spiking neural networks (SNNs) are brain-inspired models that enable energy-efficient implementation on neuromorphic hardware. However, the supervised training of SNNs remains a hard problem due to the discontinuity of the spiking neuron model.

Implicit neural networks, a.k.a., deep equilibrium networks, are a class of implicit-depth learning models where function evaluation is performed by solving a fixed

We present a discretization-free scalable framework for solving a large class of mass-conserving partial differential equations (PDEs), including the time-dependent Fokker-Planck equation and the Wasserstein gradient flow. The main observation is that the time-varying velocity field of the PDE solution needs to be self-consistent: it must satisfy a fixed-point equation involving the probability flow characterized by the same velocity field. Instead of directly minimizing the residual of the fixed-point equation with neural parameterization, we use an iterative formulation with a biased gradient estimator that bypasses significant computational obstacles with strong empirical performance. Compared to existing approaches, our method does not suffer from temporal or spatial discretization, covers a wider range of PDEs, and scales to high dimensions. Experimentally, our method recovers analytical solutions accurately when they are available and achieves superior performance in high dimensions with less training time compared to alternatives.