Ordinary differential equations (ODEs) underlie models in science and engineering, and many applications need derivatives of their solutions with respect to parameters. Ensembles of independent trajectories suit graphics processing units (GPUs), but current GPU software forces a trade-off: the fastest ensemble solvers cannot be differentiated in reverse mode at the speed they solve, and the solvers built for differentiation solve more slowly. No single tool has yet offered a reverse-mode gradient at the speed of a fused-kernel solve. We present GRADSOLVE, an open-source JAX library for solving and reverse-mode differentiating low-dimensional ODE ensembles on NVIDIA GPUs. It records the steps an adaptive solver accepts and differentiates a fixed-step replay of them; the returned gradient is the exact discrete adjoint of those steps, the same derivative Diffrax returns by default, obtained more cheaply from a fixed-length chain than from an adaptive loop. It targets ensembles differentiated many times against one recorded mesh, keeps Diffrax as a fallback, and supports explicit and Rosenbrock integrators. Used as a solver, GRADSOLVE's forward-only kernel ran 2.8x faster than DiffEqGPU.jl; used for gradients, once a record exists, it computed them 5.6-14.1x faster than Diffrax's checkpointed adjoint at matched forward-state accuracy across three GPU generations, the advantage narrowing on large ensembles and, on stiff systems, down to parity at tight accuracy. GRADSOLVE is released at https://github.com/ECLIPSE-AI4Science/gradsolve.
Sign language translation has achieved strong results with Transformer architectures, yet recent improvements largely rely on scaling model capacity at the cost of increased computation. We propose a parameter-efficient alternative that improves expressiveness without increasing model size. Rather than scaling capacity, we focus on enhancing the update dynamics of iterative refinement decoders, where each refinement step corresponds to one internal decoder iteration that progressively improves the latent representation before translation generation. We reinterpret residual refinement updates from an Ordinary Differential Equation (ODE) perspective and replace them with higher-order numerical integration schemes, namely Runge--Kutta methods (RK-2 and RK-4). These methods perform multiple function evaluations within each refinement step to produce more accurate and stable representation updates without adding decoder parameters. To the best of our knowledge, this is the first application of ODE-inspired update dynamics to sign language translation. RK-2 achieves 22.96 BLEU-4 on the PHOENIX-2014-T test set and 19.34 BLEU-4 on the CSL-Daily test set, outperforming the IPSLT baseline on both benchmarks, with fewer decoder layers and refinement iterations on CSL-Daily. These results suggest that stronger refinement dynamics can improve translation performance under parameter-efficient decoder designs, providing a complementary alternative to conventional model scaling.
Sihyeon Kim, Seunghun Lee, Vikas Singh +1cs.CV cs.LG
Diffusion and flow generative models sample by integrating a learned ODE, but high quality still requires many sequential model evaluations. Solver learning reduces this cost by adapting scalar coefficients, timesteps, or both, while keeping the backbone model fixed. In this work, we identify a structural bottleneck in this update family: each step remains span-limited. Since the scalar-coefficient update lies in the span of buffered velocity evaluations, it can fit only the in-span component while leaving any out-of-span residual unreachable by scalar recombination alone. We propose SpanLift, a lightweight neural solver that augments scalar-coefficient updates with a spatial residual operator. SpanLift keeps a fixed base solver as an in-span prior and learns a spatial residual operator over the state and velocity buffer. The operator is trained by endpoint teacher matching, preserves the pretrained backbone, and adds no model NFEs. Empirically, the learned correction transfers across base solvers and is predominantly out-of-span. Across pixel-space diffusion, latent flow matching, and precipitation nowcasting, SpanLift achieves state-of-the-art few-step sampling. With only 3 NFE, it improves CIFAR-10 FID from 8.16 to 5.69 and ImageNet FID from 17.37 to 11.83.
Filtering-based probabilistic numerical solvers for ordinary differential equations (ODEs) have been established as a flexible and efficient simulation framework with built-in numerical uncertainty quantification. However, problems that are both stiff and high-dimensional remain a challenge, as current methods are either stable and have cubic cost in the ODE dimension, or scale linearly at the expense of stability. In this paper, we close this gap and develop probabilistic ODE solvers that are both stable and scalable. We propose two complementary strategies. First, we develop a matrix-free update step that uses Jacobian-vector products, iterative linear solvers, and stochastic covariance estimation to enable linear scaling, all while retaining stability. Second, we propose iterative re-linearization to further improve stability without sacrificing scalability, turning probabilistic ODE solvers into fully implicit methods. We evaluate the proposed approaches on a range of stiff and high-dimensional problems and demonstrate improved stability and scalability over established probabilistic solvers.