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Supplementary variable method for thermodynamically consistent partial differential equations
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Data-driven identification of 2D Partial Differential Equations using extracted physical features
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Improved training of physics-informed neural networks for parabolic differential equations with sharply perturbed initial conditions
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Physics-constrained coupled neural differential equations for one dimensional blood flow modeling
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Model predictive control of nonlinear processes using neural ordinary differential equation models
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ChemNODE: A neural ordinary differential equations framework for efficient chemical kinetic solvers
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Conditional Karhunen-Loève expansion for uncertainty quantification and active learning in partial differential equation models
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Time-Accurate and highly-Stable Explicit operators for stiff differential equations
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Stabilized neural ordinary differential equations for long-time forecasting of dynamical systems
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SMS: Spiking marching scheme for efficient long time integration of differential equations
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A structurally informed data assimilation approach for nonlinear partial differential equations
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An efficient explicit implementation of a near-optimal quantum algorithm for simulating linear dissipative differential equations
We propose an efficient block-encoding technique for the implementation of the Linear Combination of Hamiltonian Simulations (LCHS) for simulating dissipative initial-value problems. This algorithm approximates a target nonunitary operator as a weighted sum of Hamiltonian evolutions, thereby emulating a dissipative problem by mixing various time scales. We introduce an efficient encoding of the LCHS into a quantum circuit based on a simple coordinate transformation that turns the dependence on the summation index into a trigonometric function. Classically, this method is equivalent to the use of a highly accurate Fejér-Clenshaw-Curtis quadrature formula. Quantumly, this significantly simplifies block-encoding of a dissipative problem and allows one to perform an exponential number of Hamiltonian simulations by a single Quantum Signal Processing (QSP) circuit. The resulting LCHS circuit has high success probability and the selector scales logarithmically with the number of terms in the LCHS sum and linearly with time. Careful analysis of error convergence proves that this method is more efficient than other LCHS circuits that have recently appeared in the literature. We verify the quantum circuit and its scaling by simulating it on a digital emulator of fault-tolerant quantum computers and, as a test problem, solve the advection-diffusion equation. The proposed algorithm can be used for simulating a wide class of nonunitary initial-value problems including the Liouville equation with added dissipation and linear embeddings of nonlinear systems, such as the Koopman-von Neumann and Carleman embeddings.