Time Without Timesteps: Simulating Coupled Dynamical Systems via Self-Consistency

Numerical simulation of dynamical systems is usually organized as a causal march through time: each state is computed from the previous one. We explore a different formulation for coupled systems. For each subsystem type we train a neural surrogate mapping a full driving trajectory and initial condition directly to a full output trajectory; following classical waveform relaxation, coupled systems are assembled by enforcing self-consistency among these trajectories: simulation becomes a fixed-point problem over complete trajectories rather than a stepwise rollout. On coupled van der Pol oscillators and Hodgkin-Huxley neuron networks, sequential depth becomes the number of solver iterations: 4-10 Newton iterations where the reference integrator takes 1500 steps. The gradient likewise loses its time recursion: it becomes a linear system solved by GMRES at memory independent of solver depth. A single scalar measured from the learned operator, the spectral radius of its Jacobian, predicts in advance where the coupled solve will converge; past that boundary, unrolled backpropagation diverges and a Neumann adjoint fails, while the implicit gradient remains correct to 0.04%. We report where the approach succeeds and where surrogate error degrades it.

Response Renormalization for Critical Deep Equilibrium Models

Deep Equilibrium Models (DEQs) compute predictions from a hidden representation unchanged by the model update. Training through this equilibrium uses implicit differentiation and requires solving an adjoint system built from the residual Jacobian. If this Jacobian is nearly singular along loss-sensitive directions, small perturbations can be strongly amplified in the adjoint response, producing large, highly sensitive gradients that can make optimization unreliable. We introduce Response Renormalization, a backward-pass framework that lifts selected near-pole denominators while leaving unlifted response channels unchanged. Collective Mode Response Renormalization (CMR) applies this correction in a low-dimensional critical subspace, while Phi-adaptive CMR computes a bounded response mass from a positive susceptibility rule. We derive dense and matrix-free collective formulations, distinguish exact gradients of a modified frozen-anchor residual from backward-response surrogates, and extend the construction to Structured Implicit Layers and Vector Attractors (SILVA). Across 23 multiphysics families spanning partial differential equations, three-dimensional fields, operator maps, complex geometries, and particle systems, CMR and Phi-CMR yield test errors no more than five percent higher than those from models trained with exact implicit differentiation in more than 98% of static and 95% of transient family-seed comparisons. Solver-index experiments show convergence toward the static adjoint, while physical-time rollouts retain predictive fidelity under the evaluated conditions. These results demonstrate that selective response renormalization can control near-critical adjoint amplification without globally damping well-conditioned sensitivity. Therefore, the method can make parameter updates more reliable while preserving the useful gradient information needed for learning.

Neural Parameter Calibration for Finite-State Mean Field Games

Mean field games efficiently approximate a very large population of strategic agents. While these games can aid the understanding of complex systems, their deployment in real-world settings is challenged by the specification of their parameters: mean field games (MFGs) often involve hidden preferences, constraints, and interactions that can rarely be theoretically derived or directly observed. To address this gap, we present a neural network-based framework for learning parametric, finite-state MFGs from observed population dynamics. To do so, we formulate the parameter calibration as an inverse problem and use implicit differentiation to backpropagate through the games' equilibrium. The resulting approach is fully differentiable and enables us to estimate flexible trajectory-wise parameter paths, including state- and time-dependent specifications without requiring observations of the individual agents' actions or rewards. We provide a proof for the exactness of the gradient computation in a discrete-time formulation. We validate our framework through numerical experiments across four systems of increasing complexity, ranging from synthetic linear-quadratic benchmarks to real-world urban mobility datasets.

Scalable Deep Unfolding of Conic Optimizers

Deep unfolding (DU) accelerates iterative optimizers by introducing learnable components and training them through unrolled iterations, but extending DU to the large-scale semidefinite programs (SDPs) common in robotics has remained limited. Unrolling a full-update conic solver such as COSMO exposes two obstacles that prior work on learned conic solvers has not: backpropagating through the per-iteration linear-system solve incurs memory quadratic in the problem size once the coefficient matrix is formed explicitly, and backpropagating through the positive semidefinite (PSD) cone projection becomes numerically unstable when eigenvalues coincide. We address the first obstacle with a matrix-free implicit differentiation rule that operates entirely through matrix-vector products, reducing memory from $O(n^2)$ to $O(n)$ and enabling backpropagation at scales where direct factorization runs out of memory. We address the second with a backward rule based on the Dalečkii--Krein representation of the Fréchet derivative, which remains well-defined under repeated eigenvalues. Together these make it possible to learn lightweight hyperparameter policies and warm-starts for a full-update conic solver. We evaluate on nonlinear covariance steering problems solved via sequential convex programming (SCP), as well as standalone SDPs and second-order cone programs ranging from max-cut and Lovász $\vartheta$ SDPs to robust estimation and control problems. The learned policies outperform state-of-the-art solvers across all problems, and can provide up to a 50$\times$ speedup depending on the class. When used as a subroutine in SCP, the learned approach delivers over a 30$\times$ speedup compared to COSMO.

Deep Unfolded Latent Optimally Partitioned-l2/l1 Networks for Data-driven Block-Sparse Recovery

The convex Latent Optimal Partition (LOP)-l2/l1 approach enables block-sparse signal recovery with unknown partitions but relies on manual hyperparameter tuning. Additionally, numerical instability in differentiating its proximal operator prevents its automatic parameter tuning via Deep Unfolding (DU). To address these limitations, we propose two architectures: a stable framework utilizing implicit differentiation and a flexible variant leveraging Deep Weight Factorization (DWF). The DWF-based approach also supports nonconvex smooth data fidelity terms. Numerical experiments demonstrate that DU-LOP-l2/l1 yields competitive performance and high resilience against impulsive noise.