Mean-field games (MFGs) study the Nash equilibrium of systems with a continuum of interacting agents, which can be formulated as the fixed-point of optimal control problems. They provide a unified framework for a variety of applications, including optimal transport (OT) and generative models. Despite their broad applicability, solving high-dimensional MFGs remains a significant challenge due to fundamental computational and analytical obstacles. In this work, we propose a particle-based deep Flow Matching (FM) method to tackle high-dimensional MFG computation. In each iteration of our proximal fixed-point scheme, particles are updated using first-order information, and a flow neural network is trained to match the velocity of the sample trajectories in a simulation-free manner. Theoretically, in the optimal control setting, we prove that our scheme converges to a stationary point sublinearly, and upgrade to linear (exponential) convergence under additional convexity assumptions. Our proof uses FM to induce an Eulerian coordinate (density-based) from a Lagrangian one (particle-based), and this also leads to certain equivalence results between the two formulations for MFGs when the Eulerian solution is sufficiently regular. Our method demonstrates promising performance on non-potential MFGs and high-dimensional OT problems cast as MFGs through a relaxed terminal-cost formulation.

The ANP (Kim et al., 2019) introduced attention into the NP family

A.1 Proof of Theorem 1 Proof log null E The first inequality is derived by Holder's inequality, so Existence is ensured as long as the chosen activation functions have at least one derivative almost everywhere. Smooth activations naturally satisfy this assumption, but it is worth noting that e.g. the ReLU activation We cannot guarantee that the Jacobian has full rank through clever choices of neural architectures. This is a natural requirement for the generator anyway. In our model, we aim to maximize the entropy of the generator, which encourages the generator to create as diverse samples as possible. In practice this ensures that the Jacobian has full rank as a degenerate Jacobian implies a reduction of entropy.

Analogy sampling based meta learning. In each subset, a training sample and a test sample can have analogy relation dependent on their problem categories. Different visual shape attributes are used among train and valid/test sets. O-IG ("in distribute four out center single"), L-R ("left center single right center single"), and U-D As shown in Table 5, our proposed method ("+Analogy") shows significantly Here, we provide details for all our models. ( r 64)

For Table 2, we use the large dataset. For Table 4 and 5, we use the small dataset. In this paper, we use fully connected neural network and convolutional neural network. For W AE, we use the MMD loss and median heuristic. For Figure 4, we use the networks of size (2-1024-1024-1) and (1-1024-1024-2) with ReLU activation functions for the encoder and decoder, respectively.

The original digits are randomly sampled from the test set. Figures are best viewed when zoomed. On FashionMNIST -bags, TargetedMI makes some mistakes among the'pullover', 'coat', and'shirt' objects (the 3rd, 5th, and 7th columns). Let us look at the bottom subfigure which depicts 10 hiragana characters and their handwriting. Figures are best viewed when zoomed.