SNAP-FM: Sparse Nonlinear Accelerated Projection for Physics-Constrained Generative Modeling

Generative models have emerged as scalable surrogates for physical simulation, yet they offer no guarantee that their outputs respect the conservation laws, boundary conditions, and nonlinear invariants that govern the underlying physics. Constrained sampling closes this gap, enforcing such constraints exactly at inference time without retraining, but at a computational cost: projection, correction, and trajectory-optimization steps are repeated during sampling, with these steps becoming expensive for nonlinear constraints. Standard ML frameworks exacerbate this: their dense tensor algebra and limited sparse solver composability obscure the structure that physical constraints naturally induce, making efficient batched nonlinear optimization difficult to realize in practice. We address this bottleneck by exploiting the structure that sample-wise batching and local PDE couplings induce in the projection subproblems -- namely, block-sparse Jacobian and KKT systems -- exposing this structure using ExaModels.jl and solving the resulting sparse nonlinear programs with MadNLP.jl and GPU sparse factorization. Applied to Physics-Constrained Flow Matching (PCFM), on PDE benchmarks with linear, nonlinear, one-dimensional, and two-dimensional constraints, this approach accelerates nonlinear constraint projection while maintaining constraint satisfaction. These results show that sparse GPU nonlinear optimization is a practical foundation for constrained generative sampling in scientific machine learning.

ABM-UDE: Developing Surrogates for Epidemic Agent-Based Models via Scientific Machine Learning

Agent-based epidemic models (ABMs) encode behavioral and policy heterogeneity but are too slow for nightly hospital planning. We develop county-ready surrogates that learn directly from exascale ABM trajectories using Universal Differential Equations (UDEs): mechanistic SEIR-family ODEs with a neural-parameterized contact rate $κ_φ(u,t)$ (no additive residual). Our contributions are threefold: we adapt multiple shooting and an observer-based prediction-error method (PEM) to stabilize identification of neural-augmented epidemiological dynamics across intervention-driven regime shifts; we enforce positivity and mass conservation and show the learned contact-rate parameterization yields a well-posed vector field; and we quantify accuracy, calibration, and compute against ABM ensembles and UDE baselines. On a representative ExaEpi scenario, PEM-UDE reduces mean MSE by 77% relative to single-shooting UDE (3.00 vs. 13.14) and by 20% relative to MS-UDE (3.75). Reliability improves in parallel: empirical coverage of ABM $10$-$90$% and $25$-$75$% bands rises from 0.68/0.43 (UDE) and 0.79/0.55 (MS-UDE) to 0.86/0.61 with PEM-UDE and 0.94/0.69 with MS+PEM-UDE, indicating calibrated uncertainty rather than overconfident fits. Inference runs in seconds on commodity CPUs (20-35 s per $\sim$90-day forecast), enabling nightly ''what-if'' sweeps on a laptop. Relative to a $\sim$100 CPU-hour ABM reference run, this yields $\sim10^{4}\times$ lower wall-clock per scenario. This closes the realism-cadence gap, supports threshold-aware decision-making (e.g., maintaining ICU occupancy $<75$%), preserves mechanistic interpretability, and enables calibrated, risk-aware scenario planning on standard institutional hardware. Beyond epidemics, the ABM$\to$UDE recipe provides a portable path to distill agent-based simulators into fast, trustworthy surrogates for other scientific domains.

End-to-End Probabilistic Framework for Learning with Hard Constraints

We present a general purpose probabilistic forecasting framework, ProbHardE2E, to learn systems that can incorporate operational/physical constraints as hard requirements. ProbHardE2E enforces hard constraints by exploiting variance information in a novel way; and thus it is also capable of performing uncertainty quantification (UQ) on the model. Our methodology uses a novel differentiable probabilistic projection layer (DPPL) that can be combined with a wide range of neural network architectures. This DPPL allows the model to learn the system in an end-to-end manner, compared to other approaches where the constraints are satisfied either through a post-processing step or at inference. In addition, ProbHardE2E can optimize a strictly proper scoring rule, without making any distributional assumptions on the target, which enables it to obtain robust distributional estimates (in contrast to existing approaches that generally optimize likelihood-based objectives, which are heavily biased by their distributional assumptions and model choices); and it can incorporate a range of non-linear constraints (increasing the power of modeling and flexibility). We apply ProbHardE2E to problems in learning partial differential equations with uncertainty estimates and to probabilistic time-series forecasting, showcasing it as a broadly applicable general setup that connects these seemingly disparate domains.

Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints

Deep generative models have recently been applied to physical systems governed by partial differential equations (PDEs), offering scalable simulation and uncertainty-aware inference. However, enforcing physical constraints, such as conservation laws (linear and nonlinear) and physical consistencies, remains challenging. Existing methods often rely on soft penalties or architectural biases that fail to guarantee hard constraints. In this work, we propose Physics-Constrained Flow Matching (PCFM), a zero-shot inference framework that enforces arbitrary nonlinear constraints in pretrained flow-based generative models. PCFM continuously guides the sampling process through physics-based corrections applied to intermediate solution states, while remaining aligned with the learned flow and satisfying physical constraints. Empirically, PCFM outperforms both unconstrained and constrained baselines on a range of PDEs, including those with shocks, discontinuities, and sharp features, while ensuring exact constraint satisfaction at the final solution. Our method provides a general framework for enforcing hard constraints in both scientific and general-purpose generative models, especially in applications where constraint satisfaction is essential.