Recent research has shown that optimization proxies can be trained to high fidelity, achieving average optimality gaps under 1% for large-scale problems. However, worst-case analyses show that there exist in-distribution queries that result in orders of magnitude higher optimality gap, making it difficult to trust the predictions in practice. This paper aims at striking a balance between classical solvers and optimization proxies in order to enable trustworthy deployments with interpretable speed-optimality tradeoffs based on a user-defined optimality threshold. To this end, the paper proposes a hybrid solver that leverages duality theory to efficiently bound the optimality gap of predictions, falling back to a classical solver for queries where optimality cannot be certified. To improve the achieved speedup of the hybrid solver, the paper proposes an alternative training procedure that combines the primal and dual proxy training. Experiments on large-scale transmission systems show that the hybrid solver is highly scalable. The proposed hybrid solver achieves speedups of over 1000x compared to a parallelized simplex-based solver while guaranteeing a maximum optimality gap of 2%.

Rapid and reliable solvers for parametric partial differential equations (PDEs) are needed in many scientific and engineering disciplines. For example, there is a growing demand for composites and architected materials with heterogeneous microstructures. Designing such materials and predicting their behavior in practical applications requires solving homogenization problems for a wide range of material parameters and microstructures. While classical numerical solvers offer reliable and accurate solutions supported by a solid theoretical foundation, their high computational costs and slow convergence remain limiting factors. As a result, scientific machine learning is emerging as a promising alternative. However, such approaches often lack guaranteed accuracy and physical consistency. This raises the question of whether it is possible to develop hybrid approaches that combine the advantages of both data-driven methods and classical solvers. To address this, we introduce UNO-CG, a hybrid solver that accelerates conjugate gradient (CG) solvers using specially designed machine-learned preconditioners, while ensuring convergence by construction. As a preconditioner, we propose Unitary Neural Operators as a modification of Fourier Neural Operators. Our method can be interpreted as a data-driven discovery of Green's functions, which are then used to accelerate iterative solvers. We evaluate UNO-CG on various homogenization problems involving heterogeneous microstructures and millions of degrees of freedom. Our results demonstrate that UNO-CG enables a substantial reduction in the number of iterations and is competitive with handcrafted preconditioners for homogenization problems that involve expert knowledge. Moreover, UNO-CG maintains strong performance across a variety of boundary conditions, where many specialized solvers are not applicable, highlighting its versatility and robustness.

A popular machine-learning model for regression tasks, including stock-market prediction, weather forecasting and real-estate pricing, is the classical support vector regression (SVR). However, a practically realisable quantum SVR remains to be formulated. We devise annealing-based algorithms, namely simulated and quantum-classical hybrid, for training two SVR models and compare their empirical performances against the SVR implementation of Python's scikit-learn package for facial-landmark detection (FLD), a particular use case for SVR. Our method is to derive a quadratic-unconstrained-binary formulation for the optimisation problem used for training a SVR model and solve this problem using annealing. Using D-Wave's hybrid solver, we construct a quantum-assisted SVR model, thereby demonstrating a slight advantage over classical models regarding FLD accuracy. Furthermore, we observe that annealing-based SVR models predict landmarks with lower variances compared to the SVR models trained by gradient-based methods. Our work is a proof-of-concept example for applying quantum-assisted SVR to a supervised-learning task with a small training dataset.

In this study, we initially investigate the application of a hybrid classical-quantum classifier (HCQC) for sentiment analysis, comparing its performance against the classical CPLEX classifier and the Transformer architecture. Our findings indicate that while the HCQC underperforms relative to the Transformer in terms of classification accuracy, but it requires significantly less time to converge to a reasonably good approximate solution. This experiment also reveals a critical bottleneck in the HCQC, whose architecture is partially undisclosed by the D-Wave property. To address this limitation, we propose a novel algorithm based on the algebraic decomposition of QUBO models, which enhances the time the quantum processing unit can allocate to problem-solving tasks.

The use of implicit time-stepping schemes for the numerical approximation of solutions to stiff nonlinear time-evolution equations brings well-known advantages including, typically, better stability behaviour and corresponding support of larger time steps, and better structure preservation properties. However, this comes at the price of having to solve a nonlinear equation at every time step of the numerical scheme. In this work, we propose a novel operator learning based hybrid Newton's method to accelerate this solution of the nonlinear time step system for stiff time-evolution nonlinear equations. We propose a targeted learning strategy which facilitates robust unsupervised learning in an offline phase and provides a highly efficient initialisation for the Newton iteration leading to consistent acceleration of Newton's method. A quantifiable rate of improvement in Newton's method achieved by improved initialisation is provided and we analyse the upper bound of the generalisation error of our unsupervised learning strategy. These theoretical results are supported by extensive numerical results, demonstrating the efficiency of our proposed neural hybrid solver both in one- and two-dimensional cases.

Large-scale numerical simulations often come at the expense of daunting computations. High-Performance Computing has enhanced the process, but adapting legacy codes to leverage parallel GPU computations remains challenging. Meanwhile, Machine Learning models can harness GPU computations effectively but often struggle with generalization and accuracy. Graph Neural Networks (GNNs), in particular, are great for learning from unstructured data like meshes but are often limited to small-scale problems. Moreover, the capabilities of the trained model usually restrict the accuracy of the data-driven solution. To benefit from both worlds, this paper introduces a novel preconditioner integrating a GNN model within a multi-level Domain Decomposition framework. The proposed GNN-based preconditioner is used to enhance the efficiency of a Krylov method, resulting in a hybrid solver that can converge with any desired level of accuracy. The efficiency of the Krylov method greatly benefits from the GNN preconditioner, which is adaptable to meshes of any size and shape, is executed on GPUs, and features a multi-level approach to enforce the scalability of the entire process. Several experiments are conducted to validate the numerical behavior of the hybrid solver, and an in-depth analysis of its performance is proposed to assess its competitiveness against a C++ legacy solver.

Direct numerical simulations (DNS) are accurate but computationally expensive for predicting materials evolution across timescales, due to the complexity of the underlying evolution equations, the nature of multiscale spatio-temporal interactions, and the need to reach long-time integration. We develop a new method that blends numerical solvers with neural operators to accelerate such simulations. This methodology is based on the integration of a community numerical solver with a U-Net neural operator, enhanced by a temporal-conditioning mechanism that enables accurate extrapolation and efficient time-to-solution predictions of the dynamics. We demonstrate the effectiveness of this framework on simulations of microstructure evolution during physical vapor deposition modeled via the phase-field method. Such simulations exhibit high spatial gradients due to the co-evolution of different material phases with simultaneous slow and fast materials dynamics. We establish accurate extrapolation of the coupled solver with up to 16.5$\times$ speed-up compared to DNS. This methodology is generalizable to a broad range of evolutionary models, from solid mechanics, to fluid dynamics, geophysics, climate, and more.

Binary Decision Diagram (BDD) based set bounds propagation is a powerful approach to solving set-constraint satisfaction problems. However, prior BDD based techniques in- cur the significant overhead of constructing and manipulating graphs during search. We present a set-constraint solver which combines BDD-based set-bounds propagators with the learning abilities of a modern SAT solver. Together with a number of improvements beyond the basic algorithm, this solver is highly competitive with existing propagation based set constraint solvers.

Systematic search and local search paradigms for combinatorial problems are generally believed to have complementary strengths. Nevertheless, attempts to combine the power of the two paradigms have had limited success, due in part to the expensive information communication overhead involved. We propose a hybrid strategy based on shared memory, ideally suited for multi-core processor architectures. This method enables continuous information exchange between two solvers without slowing down either of the two. Such a hybrid search strategy is surprisingly effective, leading to substantially better quality solutions to many challenging Maximum Satisfiability (MaxSAT) instances than what the current best exact or heuristic methods yield, and it often achieves this within seconds. This hybrid approach is naturally best suited to MaxSAT instances for which proving unsatisfiability is already hard; otherwise the method falls back to pure local search.