Aalborg and Porto Researchers Model PDEs With Tensor Networks

Researchers at the Department of Electronic Systems, Aalborg University, Aalborg, Denmark and the University of Porto, Porto, Portugal are applying concepts from quantum physics to model systems governed by advection, diffusion, reaction (ADR) partial differential equations.

The team encoded solutions to these equations as matrix product states, representing differential operators as matrix product operators, a framework for tackling these challenging problems. Time integration is performed entirely within this tensor network structure using explicit Euler updates with controlled truncation. Numerical results demonstrate the method remains compact, stable, and accurate across a range of scenarios, capturing both local solution profiles and global observables while maintaining small bond dimensions. This work highlights the potential of tensor networks as efficient structure-preserving tools for PDE simulation, and the authors acknowledge support from the Danish e-Infrastructure Consortium (DeiC) and the National Quantum Algorithm Academy (NQAA) through a postdoctoral scholarship under the project “Quantum-Driven Solutions for Multi-Agent Systems and Advanced Computation”. Advection, Diffusion, Reaction Equations and Multiscale Challenges Researchers are leveraging quantum-inspired representations to solve advection, diffusion, reaction (ADR) equations, demonstrating stable and accurate simulations. As stated in the paper, the method “shows excellent agreement with high-accuracy RK45 reference solutions and remains robust under synthetic stress tests that isolate transport, diffusion, reaction, and oscillatory initial conditions.” This approach extends to two spatial dimensions while preserving compact tensor-network representations. The researchers note that computational cost is governed primarily by the bond dimensions and tensor-network structure. “Discretized solution fields are encoded as MPS, while finite-difference derivative operators are represented as MPOs,” the study details, describing the construction of compact MPO representations of finite-difference advection and diffusion operators. This work was also partially supported by the Research Center for Systems and Technologies (SYSTEC, DOI 10.54499/UID/00147/2025) and the Associate Laboratory Advanced Production and Intelligent Systems (ARISE, DOI 10.54499/LA/P/0112/2020), both funded by Fundação para a Ciência e a Tecnologia, I.P./ MCTES through the national funds. This work builds on previous developments in quantum-inspired tensor networks, offering a more efficient approach to simulating complex phenomena.

Matrix Product States and Quantum-Inspired Tensorization The core innovation lies in representing discretized solutions as MPS, effectively encoding spatial data in a manner analogous to quantum states, while finite-difference derivative operators are translated into MPOs, allowing for streamlined calculations.

The team has implemented time integration, the process of stepping the solution forward in time, entirely within this tensor network structure. This is achieved through explicit Euler updates coupled with controlled truncation, a technique for managing computational complexity by selectively discarding less significant data. This work was also partially supported by the Research Center for Systems and Technologies (SYSTEC, DOI 10.54499/UID/00147/2025) and the Associate Laboratory Advanced Production and Intelligent Systems (ARISE, DOI 10.54499/LA/P/0112/2020), both funded by Fundação para a Ciência e a Tecnologia, I.P./ MCTES through the national funds. The researchers emphasize that their work builds on recent work demonstrating that quantum-inspired tensor networks (QTNs) can approximate the flow maps of hydrodynamic PDEs with strong compression and controlled error growth, suggesting a growing body of evidence supporting the efficacy of this approach.

The team’s approach also extends to two spatial dimensions, indicating a scalable solution for increasingly complex simulations. This method, detailed in recent work, moves beyond traditional numerical techniques by encoding discretized solutions as matrix product states (MPS) and representing derivative operators as matrix product operators (MPOs). The method is evaluated on one- and two-dimensional ADR problems and compared with high-accuracy Runge, Kutta reference solutions. Numerical results show that the proposed representation remains compact, stable, and accurate across a range of dynamical regimes.

Explicit Euler Time Stepping in Tensor-Network Form The ability to model complex systems governed by partial differential equations is fundamental to fields ranging from climate science to materials design, yet traditional computational approaches often struggle with the escalating demands of higher resolution and dimensionality. Researchers from the Department of Electronic Systems, Aalborg University, Aalborg, Denmark, and the University of Porto, Porto, Portugal, are now demonstrating that concepts borrowed from quantum physics, specifically, tensor networks, offer a compelling alternative for efficiently simulating these systems, and recent advances focus on streamlining the time-dependent solutions. This work, detailed in a recent publication, moves beyond simply representing spatial data with these networks to performing the crucial step of time integration entirely within the tensor network structure. The method is evaluated on one- and two-dimensional advection, diffusion, reaction (ADR) problems, a common challenge in modeling transport phenomena. Lower bond dimensions translate directly to reduced computational cost, which is a key indicator of efficiency.

The team compared their tensor network solver with high-accuracy Runge, Kutta solutions, finding and confirming the robustness of the approach even under challenging conditions designed to isolate specific physical effects. They suggest that many discretized PDE solution manifolds possess an “effective low-rank structure after tensorization,” allowing for simulations governed by tensor network structure rather than grid size, opening avenues for tackling increasingly complex problems in scientific computing. Low-Rank MPS Representation and Numerical Validation The expectation that complex partial differential equations demand ever-increasing computational resources is being challenged by an emerging approach leveraging concepts from quantum physics. Researchers from the Department of Electronic Systems, Aalborg University, Aalborg, Denmark, and the University of Porto, Porto, Portugal, are demonstrating that quantum-inspired tensor networks can efficiently model and solve advection, diffusion, reaction (ADR) equations, a class of equations crucial for modeling everything from pollutant dispersal to biological processes. This is not about building quantum computers to solve classical problems; rather, it’s about borrowing mathematical tools developed for quantum systems. The authors acknowledge support from the Danish e-Infrastructure Consortium (DeiC) and the National Quantum Algorithm Academy (NQAA) through a Postdoctoral Scholarship under the project “Quantum-Driven Solutions for Multi-Agent Systems and Advanced Computation,” with further funding detailed in DOI 10.54499/UID/00147/2025 and DOI 10.54499/LA/P/0112/2020. Traditional methods often rely on computationally expensive techniques, making this work significant.

The team benchmarked their tensor network solver by comparing it with high-accuracy Runge-Kutta solutions. The ability to maintain accuracy while reducing computational demands stems from the inherent low-rank structure discovered within the ADR equation’s solution manifold when expressed in this tensor format. 👉 More information🗞 Tensor Network Methods for Advection-Diffusion-Reaction Systems Using Quantum-Inspired Representations✍️ Nahid Binandeh Dehaghani, Rafal Wisniewski and A. Pedro Aguiar🧠 ArXiv: https://arxiv.org/abs/2607.14150 Stay currentSee today’s quantum computing news on Quantum Zeitgeist for the latest breakthroughs in qubits, hardware, algorithms, and industry deals. Tags: