HAMNO: A Hierarchical Adaptive Multi-scale Neural Operator with Physics-Informed Learning for Dynamical Systems
Mostafa Bamdad, Mohammad Sadegh Eshaghi, Timon Rabczuk
Abstract
Neural operators provide a powerful framework for learning solution mappings of partial differential equations directly in function space. However, many existing architectures still struggle to represent nonlinear time-dependent systems that involve multi-scale structures, long-range interactions, and stable long-time evolution. In this work, we introduce the Hierarchical Adaptive Multi-scale Neural Operator (HAMNO), a neural-operator architecture that combines local convolutional representations, global spectral operators, and hierarchical encoder-decoder processing. The central component of HAMNO is a data-dependent gating mechanism that adaptively balances local and global information at each spatial location, allowing the model to resolve fine-scale features while preserving long-range dependencies. We further develop a physics-informed extension, PI-HAMNO, based on a multi-objective loss strategy that combines data fitting with strong- and weak-form physics constraints. The strong-form term penalizes the domain-integrated squared PDE residual in physical coordinates, while the weak-form term is constructed by multiplying the governing residual by finite-element test functions and evaluating the resulting element integrals using centroid-based tetrahedral quadrature. The framework is evaluated on non-periodic Allen-Cahn (AC), Cahn-Hilliard (CH), and Swift-Hohenberg (SH) equations defined on cubic domains. Across long-horizon rollout, data-limited training, out-of-distribution initial-condition shifts, and random-seed variations, HAMNO improves predictive accuracy over standard neural-operator baselines, while PI-HAMNO further enhances stability, physical consistency, and data efficiency. The implementation is publicly available at https://github.com/MBamdad/HAMNO .
AI Impact Assessments
(1 models)Scientific Impact Assessment: HAMNO
1. Core Contribution
HAMNO introduces a neural operator architecture that combines three key components: (1) local convolutional operators, (2) global spectral (Fourier) operators, and (3) a hierarchical encoder-decoder structure, unified through a data-dependent gating mechanism that adaptively balances local and global information at each spatial location. The physics-informed extension, PI-HAMNO, adds a multi-objective loss combining strong-form (finite-difference PDE residual integrated over tetrahedra) and weak-form (finite-element Galerkin residual with P1 test functions) constraints.
The central novelty lies in the adaptive local-global fusion — rather than fixed additive combinations (as in U-FNO) or purely spectral mixing (as in FNO/U-NO), HAMNO learns spatially-varying gates α(x) and β(x) that weight local and global features. This is a meaningful architectural contribution, though the gating concept itself is well-established in deep learning (e.g., attention mechanisms, mixture-of-experts). The dual strong-weak physics loss is also a thoughtful design choice with clear justification: strong-form captures high-frequency/local PDE consistency while weak-form provides variational smoothing and naturally handles Neumann boundary conditions.
2. Methodological Rigor
Strengths in experimental design:
Concerns:
3. Potential Impact
The paper addresses a genuine practical need: accurate long-horizon prediction of multi-scale phase-field dynamics with non-periodic boundaries. Phase-field models are widely used in materials science (solidification, fracture, microstructure evolution), and accelerating these simulations has clear applied value.
The adaptive local-global gating idea could transfer to other neural operator applications where the balance between localized and long-range dynamics varies spatially and temporally. The strong-weak physics loss combination is also a reusable methodological contribution applicable beyond this specific architecture.
However, the impact may be limited by:
4. Timeliness & Relevance
Neural operators for PDE solving remain a very active area, and the specific focus on non-periodic boundary conditions and phase-field dynamics addresses real gaps. Most neural operator benchmarks use periodic problems, so this non-periodic focus is welcome. The combination of data-driven and physics-informed training is timely given increasing interest in hybrid approaches. However, the field is moving quickly toward more general architectures (geometry-aware operators, graph neural operators), and HAMNO's restriction to structured grids may limit its longevity.
5. Strengths & Limitations
Key Strengths:
Notable Weaknesses:
Overall Assessment
HAMNO represents a solid engineering contribution that thoughtfully combines established components (local convolutions, spectral operators, encoder-decoder structures, gating mechanisms) into an effective architecture for phase-field dynamics. The physics-informed extension with dual strong-weak losses is well-designed and the experimental evaluation is comprehensive within its scope. However, the novelty is somewhat incremental, the problem scale is small, and the restriction to structured cubic grids limits broader impact. The paper would benefit from larger-scale experiments, parameter-matched comparisons, and tests on more diverse geometries to fully establish its contributions.
Generated Jun 11, 2026
Comparison History (17)
Paper 2 (VideoMDM) likely has higher impact: it tackles a broadly important, timely problem (3D human motion generation) and removes the need for costly 3D ground truth by leveraging ubiquitous 2D video supervision. The diffusion-based formulation plus an expectation-equivalence result for depth-weighted reprojection provides a clear conceptual contribution with strong real-world applicability in vision/graphics/AR. It is evaluated on multiple large benchmarks and real-video datasets with competitive metrics and human preference studies. Paper 1 is technically solid but more niche to PDE surrogate modeling and incremental over existing neural-operator hybrids.
Paper 2 demonstrates a landmark achievement by exceeding the human gold-medal threshold on elite mathematical competitions (IMO and USAMO). Breakthroughs in AI mathematical reasoning and formal theorem proving have profound, multi-disciplinary implications, capturing broad attention across AI, mathematics, and computer science. While Paper 1 presents a solid methodological advancement in neural operators for physical systems, it represents an incremental progression within the specialized subfield of scientific machine learning, whereas Paper 2 signifies a major milestone in general AI capabilities.
HAMNO addresses fundamental challenges in neural operators for PDEs—multi-scale dynamics, long-range interactions, and stable long-time evolution—with broad applicability across computational physics. Its hierarchical architecture combining local convolutions, global spectral operators, and adaptive gating is a more general contribution. The physics-informed extension (PI-HAMNO) with combined strong/weak-form constraints advances the neural operator methodology significantly. While PolyFlow solves an important constrained generation problem, its scope (polytope constraints in flow matching) is narrower. HAMNO's impact spans more scientific domains involving dynamical systems and PDEs.
HAMNO addresses fundamental challenges in neural operators for PDEs—multi-scale representation, long-range interactions, and stable long-time evolution—with a principled architecture combining local/global processing and physics-informed constraints. It offers broader scientific impact across computational science, engineering, and applied mathematics. The hierarchical adaptive design with data-dependent gating is more novel architecturally, and the physics-informed extension with both strong- and weak-form constraints represents methodological rigor. Paper 1, while interesting in framing model editing as RL, is more incremental and narrower in scope with modest improvements on specific tasks.
HAMNO introduces a novel neural operator architecture with hierarchical multi-scale processing and adaptive gating, combined with a physics-informed extension using both strong and weak-form constraints. This addresses fundamental challenges in learning PDE solutions (multi-scale, long-range, stability) with broad applicability. While Paper 2 presents a useful active learning strategy for SINDy in low-data regimes, it is more incremental, building on established methods. HAMNO's architectural innovations, comprehensive evaluation across multiple equations, and publicly available implementation suggest broader impact across computational science and ML-for-science communities.
HAMNO addresses a fundamental challenge in scientific computing—learning PDE solution operators for multi-scale, nonlinear dynamical systems—with a well-structured hierarchical architecture combining local and global representations plus physics-informed constraints. Its contributions (adaptive gating, multi-objective physics loss with strong/weak forms, demonstrated improvements on challenging PDEs) have broad applicability across computational physics and engineering. Paper 1 introduces an interesting application of INRs to behavioral policy representation, but addresses a more niche problem with mixed results (amortized encoders remain competitive in many settings), limiting its broader impact.
Paper 2 (HAMNO/PI-HAMNO) likely has higher scientific impact due to broader cross-domain applicability: neural operators and physics-informed learning address PDE-driven dynamical systems across physics, engineering, climate, materials, and biology. The architecture (hierarchical multi-scale with adaptive local/global gating) plus explicit strong/weak-form constraints targets known failure modes (multi-scale, long-horizon stability, data scarcity) with methodological rigor and open-source code, supporting adoption and extensions. Paper 1 is timely and useful for LLM-agent RL, but its impact is more specialized to tool-using agents and closer to incremental refinement of credit assignment/branching.
Paper 2 addresses a critical bottleneck in scientific machine learning by effectively modeling multi-scale, non-linear PDEs. Its combination of adaptive multi-scale operators with physics-informed learning provides a rigorous, highly applicable framework for physical sciences and engineering. While Paper 1 introduces an interesting conceptual step toward graph foundation models, it remains primarily a proof-of-concept on synthetic data. Paper 2's methodological rigor, open-source availability, and immediate applicability to complex dynamical systems give it a higher potential for broad and lasting scientific impact.
Paper 2 (PAWS) likely has higher impact due to timeliness and breadth: preference-based RL is a core ingredient in modern alignment/RLHF-style systems, so fixing a fundamental training–optimization mismatch can influence many downstream methods and applications. Its contributions are conceptually focused (segment-level advantage updates), broadly applicable across tasks where human feedback is used, and directly relevant to real-world robotics and interactive AI. Paper 1 is innovative and rigorous for PDE learning, but its impact is more specialized to scientific computing/neural operators.
Paper 2 addresses a fundamental and pressing challenge in mechanistic interpretability—the reproducibility and stability of features in Sparse Autoencoders (SAEs). By demonstrating that unstable features form reproducible lower-rank subspaces, it provides deep theoretical insights and practical solutions for interpreting large neural networks. While Paper 1 offers a strong architectural advancement for scientific machine learning, Paper 2's potential to broadly impact how we evaluate and understand AI models gives it a higher overall scientific impact.