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Real-time accurate solutions of large-scale complex dynamical systems are critically needed for control, optimization, uncertainty quantification, and decision-making in practical engineering and science applications, particularly in digital twin contexts. Recent research on hybrid approaches combining numerical methods and machine learning in end-to-end training has shown significant improvements over either approach alone. However, using neural networks as surrogate models generally exhibits limitations in generalizability over different settings and in capturing the evolution of solution discontinuities. In this work, we develop a model-constrained discontinuous Galerkin Network DGNet approach, a significant extension to our previous work, for compressible Euler equations with out-of-distribution generalization. The core of DGNet is the synergy of several key strategies: (i) leveraging time integration schemes to capture temporal correlation and taking advantage of neural network speed for computation time reduction. This is the key to the temporal discretization-invariant property of DGNet; (ii) employing a model-constrained approach to ensure the learned tangent slope satisfies governing equations; (iii) utilizing a DG-inspired architecture for GNN where edges represent Riemann solver surrogate models and nodes represent volume integration correction surrogate models, enabling capturing discontinuity capability, aliasing error reduction, and mesh discretization generalizability. Such a design allows DGNet to learn the DG spatial discretization accurately; (iv) developing an input normalization strategy that allows surrogate models to generalize across different initial conditions, geometries, meshes, boundary conditions, and solution orders. In fact, the normalization is the key to spatial discretization-invariance for DGNet; and (v) incorporating a data randomization technique that not only implicitly promotes agreement between surrogate models and true numerical models up to second-order derivatives, ensuring long-term stability and prediction capacity, but also serves as a data generation engine during training, leading to enhanced generalization on unseen data. To validate the theoretical results, effectiveness, stability, and generalizability of our novel DGNet approach, we present comprehensive numerical results for 1D and 2D compressible Euler equation problems, including Sod Shock Tube, Lax Shock Tube, Isentropic Vortex, Forward Facing Step, Scramjet, Airfoil, Euler Benchmarks, Double Mach Reflection, and a Hypersonic Sphere Cone benchmark. 

Long-running scientific workflows, such as tomographic data analysis pipelines, are prone to a variety of failures, including hardware and network disruptions, as well as software errors. These failures can substantially degrade performance and increase turnaround times, particularly in large-scale, geographically distributed, and time-sensitive environments like synchrotron radiation facilities. In this work, we propose and evaluate resilience strategies aimed at mitigating the impact of failures in tomographic reconstruction workflows. Specifically, we introduce an asynchronous, non-blocking checkpointing mechanism and a dynamic load redistribution technique with lazy recovery, designed to enhance workflow reliability and minimize failure-induced overheads. These approaches facilitate progress preservation, balanced load distribution, and efficient recovery in error-prone environments. To evaluate their effectiveness, we implement a 3D tomographic reconstruction pipeline and deploy it across Argonne's leadership computing infrastructure and synchrotron facilities. Our results demonstrate that the proposed resilience techniques significantly reduce failure impact—by up to 500× —while maintaining negligible overhead (<3%). 

Deep Learning (DL), in particular deep neural networks (DNN), by default is purely data-driven and in general does not require physics. This is the strength of DL but also one of its key limitations when applied to science and engineering problems in which underlying physical properties—such as stability, conservation, and positivity—and accuracy are required. DL methods in their original forms are not capable of respecting the underlying mathematical models or achieving desired accuracy even in big-data regimes. On the other hand, many data-driven science and engineering problems, such as inverse problems, typically have limited experimental or observational data, and DL would overfit the data in this case. Leveraging information encoded in the underlying mathematical models, we argue, not only compensates missing information in low data regimes but also provides opportunities to equip DL methods with the underlying physics, and hence promoting better generalization. This paper develops a model-constrained deep learning approach and its variant TNet—a Tikhonov neural network—that are capable of learning not only information hidden in the training data but also in the underlying mathematical models to solve inverse problems governed by partial differential equations in low data regimes. We provide the constructions and some theoretical results for the proposed approaches for both linear and nonlinear inverse problems. Since TNet is designed to learn inverse solution with Tikhonov regularization, it is interpretable: in fact it recovers Tikhonov solutions for linear cases while potentially approximating Tikhonov solutions in any desired accuracy for nonlinear inverse problems. We also prove that data randomization can enhance not only the smoothness of the networks but also their generalizations. Comprehensive numerical results confirm the theoretical findings and show that with even as little as 1 training data sample for 1D deconvolution, 5 for inverse 2D heat conductivity problem, 100 for inverse initial conditions for time-dependent 2D Burgers’ equation, and 50 for inverse initial conditions for 2D Navier-Stokes equations, TNet solutions can be as accurate as Tikhonov solutions while being several orders of magnitude faster. This is possible owing to the model-constrained term, replications, and randomization.