The team, led by first author Koichi Unami, set out to close a gap they describe as both striking and difficult to ignore. Their work, published in the journal Physics of Fluids, provides a rigorous mathematical demonstration that smooth, uniform flows in steeply sloped channels are fundamentally unstable and must break up into periodic wave trains containing discontinuities. In the language of modern partial differential equations, the researchers proved that roll wave formation is not a curiosity or an artifact of imperfect models, but an unavoidable consequence of the governing equations themselves. The one-dimensional shallow water equations, a foundational model used across hydraulics, inevitably produce these discontinuous periodic patterns under the conditions typical of steep engineered channels.

Unami and his colleagues were motivated by what they saw as a persistent disconnect between the elegance of the underlying mathematics and the superficial treatment roll waves receive in civil engineering practice. “We noticed the striking gap between the mathematical and visual beauty of roll-wave phenomena, and the civil engineering community’s persistently superficial understanding of them,” Unami says. “It became difficult to ignore how confidently steady-state assumptions were repeated despite clear evidence to the contrary.” That observation framed the entire study: if the equations governing open-channel flow plainly predict unsteady, discontinuous behavior, why do design standards continue to assume that flows remain steady and uniform?

To answer that question, the researchers turned to the analytical machinery developed for hyperbolic conservation laws, the class of partial differential equations that also underlies gas dynamics and traffic flow theory. The shallow water equations describe how water depth and velocity evolve under gravity on a slope, and their solutions can develop shocks, abrupt jumps in depth and velocity, from perfectly smooth initial conditions. The team’s analysis proved the ill-posedness of uniform flows, meaning that the smooth solutions engineers often assume are mathematically unstable and cannot be trusted as physical predictions. They then characterized the resulting roll waves as weak entropy solutions, a concept central to modern fluid mechanics that allows equations to admit discontinuous solutions while still selecting the physically meaningful one through an entropy condition.

The entropy framework matters because hyperbolic equations, once shocks appear, no longer have classical solutions in the ordinary sense. Weak solutions restore mathematical meaning, but they can be non-unique, admitting spurious possibilities that violate thermodynamic common sense. Entropy conditions act as a filter, discarding unphysical solutions and isolating the ones nature actually realizes. By proving that roll waves arise as entropy solutions of the shallow water equations, the team placed these everyday rain-day phenomena within the same rigorous framework used to study shock waves in compressible gases, elevating them from empirical observations to mathematically guaranteed outcomes of the physics.

The theoretical results were reinforced by numerical experiments that simulated the evolution of flows in steep slope channels. These computations showed water flows inevitably developing periodic wave patterns marked by discontinuities, confirming that the instability of uniform flow and the emergence of roll waves are robust features rather than delicate edge cases. The combination of analytical proof and numerical verification gives the findings a dual foundation: the mathematics guarantees the outcome in principle, while the simulations demonstrate how it unfolds in practice for realistic channel conditions.

During the course of the research, conducted partly in the Middle East, the team also documented a growing regional trend: the use of roll waves on near-vertical broad channels as decorative water features. What engineers might otherwise treat as a nuisance to be suppressed has become, in architectural contexts, an aesthetic asset, with the rhythmic, self-sustaining wave trains deliberately cultivated in public fountains and landscape designs. The observation underscores how the same physical phenomenon can be a hazard in one setting and a design feature in another, depending entirely on whether it is anticipated or ignored.

That duality points to the study’s most consequential finding, a fundamental problem with existing design standards for hydraulic structures in public works. By assuming steady-state conditions and relying on numerical solutions of stable cases, such standards overlook physically relevant phenomena like roll waves entirely. Spillway chutes on dams, steeply sloped conveyance channels and similar structures are precisely the environments where the instability proven by the Kyoto-led team manifests most strongly. “Our results show that roll wave formation is not an exotic anomaly but a fundamental behavior of the governing equations,” Unami says. “Recognizing this is essential for designing hydraulic structures that reflect the realities of fluid motion, rather than idealized assumptions.”

The implications extend beyond structural safety. Roll waves carry momentum and exert unsteady forces that steady-state calculations simply cannot capture, meaning that designs validated against uniform-flow assumptions may underestimate dynamic loads, freeboard requirements and energy dissipation needs. Conversely, understanding roll wave dynamics opens opportunities for deliberate exploitation. The research team’s next ambition lies at the intersection of hydraulics and ecology: applying controlled roll wave formation to fish passages at micro-dams. Upstream-migrating fish face two competing influences in steep passages, hydrodynamic drag that hinders their movement and rheotactic stimulation, the flow-based cues that orient fish upstream. The researchers hope that tuning roll wave formation could balance these forces, easing passage while maintaining the flow conditions fish respond to naturally.

Such an application would require integrating fluid mechanics with ecological design, a synthesis the team regards as the natural continuation of their work. The broader lesson of the study is methodological as much as mathematical: phenomena that appear as minor irregularities in field observations can, when examined through the rigorous lens of partial differential equations, reveal themselves as essential, inevitable behaviors of the system. Roll waves have been visible on rainy roads for as long as there have been rainy roads, but only now has their formation been formally established as a mathematical certainty of the equations that govern shallow water flow. For hydraulic engineers, the message is that the waves rolling down a spillway chute are not noise to be averaged away, but signal, a fundamental feature of fluid motion that design standards must finally take into account.

Subject of Research: Mathematical analysis of roll wave formation and instability of uniform flows in the shallow water equations

Article Title: Rolling with the waves

Article References: Rolling with the waves. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: roll waves, shallow water equations, hydraulic engineering, weak entropy solutions, flow instability, dam spillways, partial differential equations, fluid mechanics, open-channel flow, fish passages, Kyoto University, discontinuities

A team led by Casey E. Stowers and Thomas E. Yankeelov at the University of Texas at Austin and MD Anderson Cancer Center has now systematically tested how best to represent tumor cell movement in these predictive models. Their work, published in the Annals of Biomedical Engineering, draws on data from 141 patients with locally advanced triple-negative breast cancer enrolled in the ARTEMIS clinical trial at MD Anderson. Each patient received four cycles of Adriamycin and Cytoxan, with magnetic resonance imaging scans taken before treatment, after two cycles, and after four cycles. Patients whose tumors shrank by at least 70 percent on ultrasound went on to receive twelve cycles of Taxol, while others received experimental therapies according to trial guidelines. After the second course of therapy, all patients underwent surgery, at which point their pathological complete response status was determined.

The modeling framework rests on partial differential equations that track the number of tumor cells in every three-dimensional voxel of the breast over time. The number of cells in each voxel is estimated from diffusion-weighted MRI using an inverse relationship between the apparent diffusion coefficient of water in tissue and tumor cell density. The team compared four increasingly sophisticated models. The simplest, a reaction-only model, describes logistic proliferation of cells up to a carrying capacity and drug-induced cell death driven by a concentration map derived from dynamic contrast-enhanced MRI. The second adds a diffusion term, capturing cells migrating from regions of high density to low density. The third mechanically couples that diffusion term, damping the diffusion coefficient exponentially via the von Mises stress computed from a linear elastic mechanical model of the breast, reflecting the experimental observation that solid stress inhibits tumor cell invasion. The fourth and most complex model adds an advection term, in which cells are transported by a velocity field calculated as the time derivative of the mechanical displacement field—an approach originally developed for modeling glioma growth in the brain.

The rationale for exploring advection is subtle but important. In triple-negative breast cancer, tumors typically shrink during therapy rather than expand, and a diffusion term alone cannot physically represent the inward movement of cells that accompanies shrinkage. Advection, by contrast, can carry cells toward the tumor center as the mass effect diminishes with treatment-induced cell death. In the two example patients highlighted in the study—one who achieved a complete response and one who did not—the velocity maps revealed striking behavior. On days when the total tumor cell count was rising, the modeled velocities pointed outward, in the direction of growth. On days shortly after drug delivery, when cell counts were falling, the velocities reversed and pointed toward the tumor center, pulling cells inward and producing smaller, denser tumor maps that more closely matched the imaging data.

That mechanistic insight translated into measurable differences in model performance. For the patient who achieved a complete response, the advection model produced a concordance correlation coefficient of 0.44 at the calibration timepoint compared with 0.31 for the three simpler models, and a Dice score of 0.65 versus 0.58, indicating substantially better spatial overlap with the measured tumor. The advection model also predicted far fewer disconnected islands of tumor cells—two subvolumes instead of seven at the second imaging visit—producing tumor geometries that looked more biologically realistic. Similar patterns held for the non-responding patient, whose residual tumor was better captured by the advection model than by any of the alternatives.

Across the full cohort of 141 patients, however, the picture became more nuanced. At the calibration timepoint, the reaction–mechanically coupled diffusion–advection model was significantly more accurate than all three simpler models on every global and local metric examined, including the absolute difference in percent change of total tumor cell count and volume, voxel-wise mean square error, and concordance correlation coefficient. But at the prediction timepoint—the third MRI scan, which the models had never seen—no significant differences emerged among the four models. The median absolute difference between predicted and measured percent change in tumor volume across the cohort was 9.1 percent for both the reaction-only and reaction–diffusion models, and the advection model offered only a small, statistically insignificant improvement of roughly 4 to 5 percent in tumor volume error for the non-responding patients, who are precisely the group clinicians most need to forecast accurately.

The receiver operating characteristic analysis reinforced this equivalence. When the researchers used predicted total tumor cell count or tumor volume at the third scan to classify patients by their eventual pathological complete response status among the 102 patients who received Taxol as their second therapy, no significant differences appeared between the areas under the ROC curves for any of the four models—or, notably, between the model predictions and the measured imaging data themselves. In other words, even the simplest model predicted complete response status as well as the actual measurements did, and all four models achieved comparable accuracy in this clinically critical task.

Computational cost, however, separated the models dramatically. The reaction-only model calibrated in a median of about 0.31 minutes per patient, while the reaction–diffusion model took roughly 118 minutes, and the mechanically coupled diffusion and advection models required around 222 to 252 minutes each, because both demand a full mechanical equilibrium solve at every optimization iteration. Each calibration ran on a 24-core server, and the two calibrated parameters—global tumor cell proliferation rate and global drug efficacy—were identical across all four models, fitted using the Levenberg–Marquardt algorithm against the first two imaging visits. The authors suggest that reduced-order modeling techniques and higher-order numerical schemes could eventually narrow this efficiency gap for the most accurate model.

One of the study’s most consequential findings concerns the diffusion term itself. Adding diffusion, whether mechanically coupled or not, provided no measurable improvement in calibration or prediction accuracy compared with the reaction-only model. The researchers attribute this to the insensitivity of the diffusion coefficient in this disease setting, where tumors mostly shrink rather than expand; prior sensitivity analyses found the parameter’s influence fell below the threshold for importance even when varied over five orders of magnitude. Yet the authors caution against simply deleting the term: a reaction-only model cannot represent tumor invasion or any expansion in tumor volume, which would produce dangerously poor predictions for a patient whose disease progresses during therapy. Their recommendation is that, among models without advection, the reaction–diffusion model offers the best balance, capturing invasion while calibrating far faster than the mechanically coupled variant.

The implications extend well beyond breast cancer. Because the advection term was adapted from brain tumor modeling, and because the conclusions here apply to tumors that largely shrink under treatment, the framework could inform computational oncology across many solid tumor types. The authors outline several avenues for improvement: incorporating the Allee effect into the proliferation term to better capture the death of small tumors, subdividing tumor cells into sensitive and resistant populations, coupling hydrostatic stress to proliferation or drug resistance, and replacing the linear elastic tissue assumption with nonlinear hyperelastic or anisotropic material models. They also note that future validation should draw on multi-site datasets such as the I-SPY 2 trial, which spans multiple breast cancer subtypes. For now, the study delivers a clear practical message: the reaction term carries most of the predictive power for shrinking tumors, advection is the only term that can physically capture inward cell movement during treatment-induced shrinkage, and choosing among these formulations requires weighing calibration speed against the risk of missing a growing tumor.

Subject of Research: Partial differential equation modeling of tumor cell movement in triple-negative breast cancer receiving neoadjuvant therapy

Article Title: Characterizing Tumor Cell Movement in Partial Differential Equation Models of Triple-Negative Breast Cancer Receiving Neoadjuvant Therapy

Article References: Stowers, C. E., Wu, C., Lorenzo, G., Hormuth, D. A., II, Yam, C., Ma, J., Rauch, G. M., & Yankeelov, T. E. (2026). Characterizing Tumor Cell Movement in Partial Differential Equation Models of Triple-Negative Breast Cancer Receiving Neoadjuvant Therapy. Annals of Biomedical Engineering. https://doi.org/10.1007/s10439-026-04348-7

Image Credits: AI Generated

DOI: 10.1007/s10439-026-04348-7

Keywords: triple-negative breast cancer, neoadjuvant therapy, reaction-diffusion-advection, mathematical oncology, MRI, tumor forecasting, model calibration, advection, mechanical coupling, pathological complete response, partial differential equations, computational biology

The review, led by Mohammad Nooraiepour of the University of Oslo with colleagues from the University of Bergen, Siemens Energy AI Lab, and Friedrich-Schiller-University Jena, does not declare a winner in the contest between traditional solvers and neural networks. Instead, it delivers a more unsettling and more useful conclusion: the two paradigms are epistemologically different kinds of machinery, and confusing them leads to both overhyped claims about artificial intelligence and underuse of what it genuinely offers. Classical methods are deductive, meaning their errors can be bounded by quantities derived directly from the structure of the equation and the parameters of the discretization. Machine learning methods are inductive, meaning their accuracy depends on statistical proximity to the data they were trained on. That single distinction, the authors argue, determines not which approach is superior but what each can certify, and therefore where each should be deployed.

To make the comparison rigorous, the review organizes the entire field around six fundamental computational challenges: high dimensionality, nonlinearity, geometric complexity, discontinuities, multiscale phenomena, and multiphysics coupling. These six axes function as a common testing ground. Classical methods, including finite difference, finite element, finite volume, and spectral discretizations, are assessed on each axis for their structure-preserving properties, their rigorous convergence theory, and their scalable solver design. The verdict is nuanced. On problems of moderate dimension with well-behaved geometry, classical methods remain unmatched, delivering certified accuracy with error estimates that engineers can take to a regulator or a certification authority. But the review is candid about their persistent weaknesses: the curse of dimensionality crushes grid-based methods as the number of variables grows, and geometrically complex domains can make mesh generation a bottleneck that consumes more human effort than the solve itself.

Machine learning approaches are then subjected to exactly the same critical assessment, which is itself a departure from much of the existing literature, where neural methods are typically benchmarked only against other neural methods. The review builds a taxonomy organized by the degree to which physical knowledge is incorporated into the learning process. At one end sit purely data-driven surrogates; in the middle sit physics-informed neural networks, which embed the residual of the governing PDE directly into the loss function so that the network is penalized for violating the physics; and further along sit neural operators, which learn mappings between function spaces rather than pointwise values, promising generalization across families of related problems. Graph-based architectures, transformers, and generative models round out the landscape, alongside hybrid frameworks that interleave classical solvers with learned components.

The critical findings here are sobering for anyone hoping that neural networks will simply replace finite element solvers. Physics-informed neural networks, despite enormous popularity, struggle with stiff problems, sharp gradients, and the kind of discontinuous behavior that dominates real multiphysics applications such as fracture propagation or salt precipitation during carbon dioxide storage. Their training is often fragile, their convergence theory is far less mature, and their accuracy guarantees are fundamentally statistical rather than deterministic. Neural operators fare better on generalization across parameter ranges, but the review emphasizes that their reliability is always conditional on the training distribution. A neural surrogate that performs beautifully within the range of parameters it has seen can fail catastrophically, and silently, the moment it is asked to extrapolate. Classical methods never fail silently in quite the same way, because their error bounds degrade predictably with refinement.

What elevates the review above a simple scorecard is its treatment of complementarity. The authors identify three genuine synergies between the paradigms. First, machine learning can accelerate the inner workings of classical solvers, for example by learning preconditioners, coarse-grid corrections, or adaptive mesh indicators, while the outer classical framework retains its convergence guarantees. Second, classical methods can generate the high-fidelity training data and validation benchmarks that neural surrogates require, creating a pipeline in which the deductive method certifies the inductive one. Third, learned models can capture constitutive behavior or subgrid physics that is too expensive or too poorly understood to resolve explicitly, feeding these closures into structure-preserving classical schemes. In each case, the division of labor follows the epistemological split: the learned component supplies speed and flexibility where statistics suffice, and the classical component supplies certification where proof is required.

The review goes further and develops design principles for hybrid methods, including a framework for what the authors call the structure inheritance problem, that is, how to ensure that a learned component does not destroy the conservation laws, stability properties, or symmetry structures that a classical discretization was carefully constructed to preserve. They also propose an error budget decomposition that separates the total error of a hybrid simulation into three contributions: the discretization error of the classical scheme, the approximation error of the neural component, and the coupling error introduced at their interface. This decomposition matters because it turns hybrid design from an art into an accounting exercise. If a coupled simulation goes wrong, the framework tells practitioners where to look, and it prevents the common failure mode in which a neural component’s unquantified error is silently absorbed into an apparently converged classical result.

Looking forward, the authors assess emerging frontiers with the same skeptical lens. Foundation models for scientific computing, inspired by the large language models that transformed natural language processing, are evaluated against the structural constraints of PDE solving, and the review asks whether the scaling laws that powered language models can plausibly transfer to problems governed by conservation laws and boundary conditions. Differentiable programming for inverse design, in which entire simulation pipelines are made end-to-end differentiable so that gradients can flow from design objectives back through the solver, is judged promising but dependent on the same error-accounting discipline. Quantum algorithms for PDEs are treated with particular caution: the review distinguishes between barriers that are fundamental and those that are merely contingent on engineering progress, and finds that for several quantum approaches the current obstacles look structural rather than incidental. Exascale co-design, the joint development of algorithms and hardware at extreme scale, is identified as a domain where hybrid classical-learned solvers may find their most immediate practical impact.

The timing of this synthesis is significant. Scientific machine learning has grown from a niche curiosity into a field with its own conferences, journals, and funding programs, and the hype cycle has produced both genuine breakthroughs and inflated claims. Reviews that merely catalog methods multiply yearly, but rigorous evaluations that apply identical criteria to both paradigms remain rare. By insisting that epistemological character, rather than computational speed, is the primary criterion for responsible method selection, the review provides something the community has lacked: a principled basis for deciding, problem by problem, whether a learned surrogate, a certified classical solver, or a carefully engineered hybrid is the right tool. The stakes are concrete. Applications such as geological carbon storage, where the authors’ own research is anchored through projects on salt precipitation and convective mixing during CO2 injection, demand simulations that are simultaneously fast enough for uncertainty quantification over thousands of realizations and trustworthy enough to inform regulatory decisions about subsurface storage integrity.

The conclusion the authors reach is not that one paradigm supersedes the other, but that their principled integration offers the most credible path toward computationally tractable solutions to the most consequential open problems that neither approach can address alone. For decades, computational science advanced by refining discretizations, solvers, and hardware within a single deductive tradition. The last several years added a second tradition with fundamentally different guarantees. The review’s lasting contribution may be its refusal to choose between them, and its demonstration that the future of simulation lies in knowing precisely what each tradition can promise, what it cannot, and how to combine them without letting the strengths of one conceal the weaknesses of the other. In an era when artificial intelligence is routinely described as a replacement for existing tools, this is a refreshingly precise message: the equations that govern the physical world will be solved not by machines alone, nor by classical mathematics alone, but by an alliance whose terms are now, finally, being written down.

Subject of Research: A critical synthesis of classical numerical, machine learning, and hybrid methods for solving partial differential equations

Article Title: Partial differential equations in the age of machine learning: a critical synthesis of classical, machine learning, and hybrid methods

Article References: Nooraiepour, M., Both, J. W., Kadeethum, T., & Sadeghnejad, S. (2026). Partial differential equations in the age of machine learning: a critical synthesis of classical, machine learning, and hybrid methods. Artificial Intelligence Review. https://doi.org/10.1007/s10462-026-11677-5

Image Credits: AI Generated

DOI: 10.1007/s10462-026-11677-5

Keywords: partial differential equations, machine learning, physics-informed neural networks, neural operators, finite element methods, hybrid computational methods, scientific machine learning, structure-preserving discretization, error quantification, multiphysics simulation, foundation models, carbon dioxide storage

Neural operators have quietly become one of the most consequential tools in computational science. Instead of learning a mapping between finite vectors, they learn mappings between functions, which makes them natural solvers for partial differential equations. Given a family of PDEs describing, say, fluid flow through porous rock or heat diffusing through a material, a trained neural operator can predict the full solution field for a new set of parameters almost instantly, bypassing the expensive numerical solvers that would otherwise be required. Classical architectures such as DeepONet and the Fourier neural operator have transformed this landscape, but their quantum counterparts have lagged behind, hampered by processing overheads and theoretical gaps about what quantum circuits can actually represent.

The central obstacle has been scaling. Many quantum machine-learning paradigms demand qubit counts or circuit depths that grow so quickly with problem size that they collapse into impracticality on near-term hardware. The Shanghai team, led by Ruocheng Wang, Xiaoqiu Zhong, Zhuo Xia and Junchi Yan, attacked the problem from two directions at once: they built a leaner architecture and, crucially, they proved something rigorous about its power.

The theoretical centerpiece is an extension of the universal approximation theorem to the quantum domain for continuous nonlinear operators. Universal approximation, first established for classical neural networks in the 1990s, guarantees that sufficiently wide networks can represent a broad class of functions; the operator version underpins DeepONet. Proving an analogous guarantee for quantum circuits closes a foundational gap, but the team went further. Departing from the conventional view that quantum advantage must flow from the exponential size of Hilbert space, they showed that the architecture’s density matrix implicitly constructs a quadratic feature frame.

That quadratic frame yields a striking expressivity bound. For operators, the implicit feature space scales as O(p²) in the number of parameters p, circumventing the O(p) linear capacity limits of matched classical models. In plain terms, each additional parameter in the quantum model buys roughly a square’s worth of representational capacity compared with a classical model of the same size. This is a subtle but meaningful advantage: it does not rely on exotic claims about exponentially large state spaces, but on a concrete, provable property of how the quantum circuit encodes features.

The second innovation addresses a practical bottleneck known as spectrum alignment. Quantum models encode input data through frequency modulation, and capturing high-frequency components of a solution, the sharp gradients and fine oscillations that matter in real PDEs, typically requires either deep circuits or many parameters. The researchers introduced a trainable-frequency strategy, dubbed TF-QuanONet, in which the base frequencies of the encoding adaptively space themselves during training. The network effectively learns which frequencies to emphasize, capturing relatively high-frequency structure without inflating the parameter count or the circuit depth.

The benchmarks are where the claims meet reality. Across extensive experiments, TF-QuanONet notably outperformed competing quantum baselines and achieved accuracy competitive with classical frameworks under strictly matched-parameter conditions, a fairness constraint that quantum machine-learning comparisons often fail to honor. More intriguing still is what happened as the problems grew. In high-dimensional scaling regimes, with latent dimensions p approaching 256, the quantum architecture exhibited superior optimization robustness, consistently converging to its intrinsic error floor while classical baselines suffered from high variance. The quantum model was not just accurate; it was reliably trainable where its classical competitors became erratic.

The team also tested the architecture on real IBM quantum processors, including the ibm_fez device, as a qualitative proof of concept. Comparisons between noise-free simulations and physical hardware demonstrated that QuanONet retains functional resilience on near-term machines, a nontrivial achievement given that real qubits decohere, gates misfire, and measurement noise corrupts outputs. The experiments spanned a range of canonical problems, including dynamical systems, advection equations and Darcy flow, with visualizations confirming that predictions track ground-truth solutions across varying input frequencies.

The work arrives amid a broader reckoning in quantum machine learning, where researchers have grown wary of claims that evaporate under fair comparison or realistic hardware assumptions. By grounding its architecture in a proven expressivity theorem, keeping resource requirements modest, and validating on commercial hardware, the study offers a template for what credible quantum advantage in scientific machine learning might look like. The code has been released publicly on GitHub and archived on Zenodo, and all datasets were generated directly from it, inviting the community to scrutinize and extend the results.

Whether the quadratic expressivity edge translates into decisive practical wins as quantum hardware improves remains an open question, but the study reframes the debate. Rather than waiting for fault-tolerant machines or betting everything on exponential Hilbert spaces, it demonstrates that carefully designed quantum architectures, with theory and engineering aligned, can already hold their own against strong classical baselines on the noisy processors available today.

Subject of Research: A quantum neural operator architecture with a proven quadratic expressivity bound for solving partial differential equations on near-term quantum hardware.

Article Title: Quantum neural operators with implicit quadratic frame and expressivity advantages

Article References: Wang, R., Zhong, X., Xia, Z., & Yan, J. (2026). Quantum neural operators with implicit quadratic frame and expressivity advantages. Nature Machine Intelligence. https://doi.org/10.1038/s42256-026-01289-7

Image Credits: AI Generated

DOI: 10.1038/s42256-026-01289-7

Keywords: quantum machine learning, neural operators, partial differential equations, QuanONet, NISQ era, universal approximation, expressivity, trainable frequencies, IBM quantum processors, DeepONet, quantum computing, scientific machine learning