Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Neural Variable-Order Fractional Differential Equation Networks
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes NvoFDE, a neural network framework whose fractional derivative order is a learnable function of time and hidden state, instantiated as variable-order FROND GNNs (Nvo-GRAND, Nvo-CDE) and as a PINN-style solver for variable-order FDEs.
Problem
Constant-order fractional neural differential equations cannot flexibly model memory that varies with time or hidden features; variable-order FDEs are rarely combined with learnable neural networks.
Approach
Let the fractional derivative order alpha(t, x(t)) be a learnable neural function of time and hidden features (constrained to (0,1]), generalizing neural ODEs and constant-order neural FDEs, and apply it to FROND-style graph dynamics.
Key results
- Nvo-CDE reaches 93.42% on Roman-empire and 98.53% on Minesweeper versus 96.04% for F-CDE on Minesweeper, a reported 2.49% gain from variable-order derivatives on that dataset.
- Nvo-GRAND-l achieves 86.0% on Cora and 97.4% on Disease, above F-GRAND-l on most homophilic datasets.
- On the Verhulst-Pearl equation, test loss reaches about 3.3e-6 after 2000 iterations with 10 to 50 discretization points.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use variable-order Nvo-CDE instead of constant-order F-CDE | Nvo-CDE · Minesweeper | ROC-AUC: 96.04$\pm$0.25 -> 98.53$\pm$0.27; 2.49% | Helped |
| Use variable-order Nvo-CDE instead of constant-order F-CDE | Nvo-CDE · Roman-empire | accuracy: 93.06$\pm$0.55 -> 93.42$\pm$0.22 | Helped |
| Use variable-order Nvo-CDE instead of constant-order F-CDE | Nvo-CDE · Wiki-cooc | accuracy: 98.73$\pm$0.68 -> 99.32$\pm$0.28 | Helped |
| Use variable-order Nvo-CDE instead of constant-order F-CDE | Nvo-CDE · Questions | ROC-AUC: 75.17$\pm$0.99 -> 74.87$\pm$0.23 | Hurt |
| Use variable-order Nvo-CDE instead of constant-order F-CDE | Nvo-CDE · Workers | ROC-AUC: 82.68$\pm$0.86 -> 83.33$\pm$0.65 | Helped |
| Use variable-order Nvo-CDE instead of constant-order F-CDE | Nvo-CDE · Amazon-ratings | accuracy: 49.01$\pm$0.56 -> 50.09$\pm$0.40 | Helped |
| Use variable-order Nvo-GRAND-l instead of constant-order F-GRAND-l | Nvo-GRAND-l · Cora | accuracy: 84.8$\pm$1.1 -> 86.0\(\pm\)0.5 | Helped |
| Use variable-order Nvo-GRAND-l instead of constant-order F-GRAND-l | Nvo-GRAND-l · Computer | accuracy: 84.4$\pm$1.5 -> 87.9\(\pm\)0.8 | Helped |
| Use variable-order Nvo-GRAND-l instead of constant-order F-GRAND-l | Nvo-GRAND-l · Photo | accuracy: 92.8$\pm$0.6 -> 94.1\(\pm\)0.2 | Helped |
| Use variable-order Nvo-GRAND-l instead of constant-order F-GRAND-l | Nvo-GRAND-l · CoauthorPhy | accuracy: 94.5$\pm$0.4 -> 94.7\(\pm\)0.2 | Helped |
| Use variable-order Nvo-GRAND-nl instead of constant-order F-GRAND-nl | Nvo-GRAND-nl · ogbn-arxiv | accuracy: 71.4$\pm$0.3 -> 72.0\(\pm\)0.2 | Helped |
| Use variable-order Nvo-GRAND-nl instead of constant-order F-GRAND-nl | Nvo-GRAND-nl · Disease | accuracy: 85.5$\pm$2.5 -> 89.8\(\pm\)3.4 | Helped |
Limitations
- Learned order alpha(t,x) restricted to (0,1], so higher-order dynamics must be converted; gains on some datasets are small or absent (e.g. Questions, Amazon-ratings differences within noise).
- Image classification results are only marginal improvements and are deferred to the appendix.
- Theoretical treatment of variable-order differential vs integral forms is noted as non-trivial and not fully resolved.
Experimental setup
Datasets: Cora Citeseer Pubmed CoauthorCS Computer Photo CoauthorPhy ogbn-arxiv Airport Disease Roman-empire Wiki-cooc
Benchmarks: Cora Citeseer Pubmed CoauthorCS Computer Photo CoauthorPhy ogbn-arxiv Airport Disease Roman-empire Wiki-cooc
Metrics: accuracy ROC-AUC test loss
Baselines: GCN GAT HGCN GIL GRAND-l GRAND-nl F-GRAND-l F-GRAND-nl GraphBel NSD CDE F-CDE
Builds on: FROND GRAND CDE (continuous graph dynamics) Neural ODE PINN
Closest papers by meaning
- Efficient Training of Neural Fractional-Order Differential Equation via Adjoint Backpropagation (AAAI 2025 · similarity 0.90)
- Coupling Graph Neural Networks with Fractional Order Continuous Dynamics: A Robustness Study (AAAI 2024 · similarity 0.87)
- Distributed-Order Fractional Graph Operating Network (NeurIPS 2024 · similarity 0.87)
- Unleashing the Potential of Fractional Calculus in Graph Neural Networks with FROND (ICLR 2024 · similarity 0.86)
- Operator-Learning-Inspired Modeling of Neural Ordinary Differential Equations (AAAI 2024 · similarity 0.84)
- On the Generalization and Approximation Capacities of Neural Controlled Differential Equations (ICLR 2024 · similarity 0.84)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.