ReLU-KAN: New Kolmogorov-Arnold Networks that Only Need Matrix Addition, Dot Multiplication, and ReLU
Abstract: Limited by the complexity of basis function (B-spline) calculations, Kolmogorov-Arnold Networks (KAN) suffer from restricted parallel computing capability on GPUs. This paper proposes a novel ReLU-KAN implementation that inherits the core idea of KAN. By adopting ReLU (Rectified Linear Unit) and point-wise multiplication, we simplify the design of KAN's basis function and optimize the computation process for efficient CUDA computing. The proposed ReLU-KAN architecture can be readily implemented on existing deep learning frameworks (e.g., PyTorch) for both inference and training. Experimental results demonstrate that ReLU-KAN achieves a 20x speedup compared to traditional KAN with 4-layer networks. Furthermore, ReLU-KAN exhibits a more stable training process with superior fitting ability while preserving the "catastrophic forgetting avoidance" property of KAN. You can get the code in https://github.com/quiqi/relu_kan
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What is this paper about?
This paper introduces a new kind of neural network layer called ReLU-KAN. It’s a faster, simpler version of Kolmogorov–Arnold Networks (KANs), a type of model designed to learn complicated relationships by adding together many simple “bump-shaped” functions. ReLU-KAN replaces the complex math in KANs with easy-to-compute pieces that work very well on modern computer chips (GPUs), making training much faster and more stable.
What questions did the researchers ask?
The researchers explored three main questions:
- Can we replace KAN’s complicated building blocks (called B-splines) with something much simpler that computers can compute quickly?
- If we do, will the new version still learn as accurately—or even better—than the original KAN?
- Will the new version keep a special strength of KANs: learning new things without forgetting what they already learned (called “catastrophic forgetting avoidance”)?
How did they do it?
To explain their approach, think of learning a shape by stacking small “bumps”:
- In classic KANs, each bump is made from a complex curve called a B-spline. These are powerful but slow and hard to compute in parallel.
- The authors swapped those out for simpler bumps built from ReLU, a basic function used in deep learning that keeps positive numbers and turns negatives into zero. You can think of ReLU as a “no negatives allowed” filter.
Here’s the idea in everyday terms:
- Each new bump is created by multiplying two ReLUs that act like a window: one keeps values below a top edge, the other keeps values above a bottom edge. Multiplying them shapes a tent-like bump that’s nonzero only in a small interval.
- They square the bump to make it smoother and normalize it so its size is consistent. This creates a clean, bell-shaped bump that’s cheap to compute.
Instead of handling each bump one by one, they:
- Pre-draw grid lines (start and end points for each bump) so the computer doesn’t have to re-figure them every time.
- Turn the whole process into matrix operations (like doing math on entire tables of numbers at once), which GPUs are excellent at.
- Use a “convolution” step (imagine sliding a small stencil over a row of numbers to sum weighted pieces) to combine all bumps efficiently.
All of this is easy to implement in popular tools like PyTorch—in fact, the core layer fits in under 30 lines of code.
What did they find?
The researchers tested ReLU-KAN and the original KAN on several math functions and measured speed, accuracy, and training stability. Here’s what they found:
- Much faster training:
- ReLU-KAN trained about 5× to 20× faster than KAN, especially on GPUs and as the models got bigger.
- More accurate fitting:
- ReLU-KAN often achieved much lower error—up to about 100× smaller in some cases—meaning it matched the target functions more closely.
- More stable learning:
- The training process had smoother, more reliable progress (fewer ups and downs in the loss curve).
- Keeps the “don’t forget old stuff” benefit:
- Like KANs, ReLU-KAN avoided “catastrophic forgetting” in tests where the model learned different parts of a problem in phases.
Why does this matter?
ReLU-KAN shows that:
- You can keep the big idea behind KANs—building functions from simple one-dimensional bumps—while making it fast and easy to run on modern hardware.
- Faster and more accurate models mean researchers and engineers can train bigger or more complex systems in less time.
- Because the method uses standard operations (matrix math, ReLU, convolution), it slots right into existing deep learning frameworks and could be combined with popular architectures like CNNs or Transformers.
- The ability to learn new tasks without forgetting old ones makes ReLU-KAN promising for continual learning—useful in robotics, personalization, and systems that update over time.
In short, ReLU-KAN keeps what’s special about KANs but makes it practical: simpler math, faster training, better accuracy, and strong memory of what it has learned.
Practical Applications
Overview
The paper proposes ReLU-KAN, a GPU-friendly reimplementation of Kolmogorov-Arnold Networks (KAN) that replaces B-spline basis functions with a simple, normalized ReLU-based basis and expresses all computations as matrix operations and convolutions. This enables 5–20× faster training, more stable convergence, and higher fitting accuracy while preserving KAN’s resilience to catastrophic forgetting. The implementation is compact (≈30 lines of PyTorch), relies only on ReLU, matrix addition, dot products, and convolution, and precomputes non-trainable parameters for efficiency.
Below are practical applications derived from these findings, organized into immediate and long-term categories. Each item highlights relevant sectors, potential tools/products/workflows, and assumptions or dependencies affecting feasibility.
Immediate Applications
These applications can be deployed now using the provided PyTorch implementation and standard GPU infrastructure.
- Faster function approximation and regression pipelines
- Sectors: software/ML engineering, scientific computing, manufacturing, R&D
- Tools/products/workflows:
- Drop-in ReLU-KAN layer for PyTorch models to replace or complement KAN/MLP blocks for scalar/regression tasks
- ONNX/TensorRT export (ops are matmul/conv/ReLU) for production inference
- Hyperparameter workflow: tune number of grids G and span k to trade off capacity/locality
- Assumptions/dependencies:
- Reported gains are on function-approximation benchmarks; validate on domain-specific datasets
- Memory/time trade-offs depend on G, k, and input dimensionality
- Continual/online learning with reduced catastrophic forgetting
- Sectors: robotics (adaptive control), AIOps (concept drift), IoT analytics, personalization
- Tools/products/workflows:
- Online update loop where basis weights are adapted per task segment without replay buffers
- Safety-critical workflows: gradual adaptation in changing environments (e.g., robot on new surfaces)
- Assumptions/dependencies:
- Catastrophic forgetting results are demonstrated on synthetic 1D tasks; behavior on large-scale, multi-task settings must be evaluated
- Requires monitoring and early stopping to avoid drift in unstable regimes
- Time-series modeling with GPU-efficient KAN-like layers
- Sectors: finance (forecasting), supply chain (demand), telecom (traffic), energy (load/renewable output), healthcare (vital signs)
- Tools/products/workflows:
- Replacing MLP heads in TKAN-style time-series models with ReLU-KAN for faster training and stable fitting
- Real-time forecasting services using Jetson/edge GPUs due to low-latency ops
- Assumptions/dependencies:
- Validate against strong baselines (e.g., Temporal Convolutional Networks, Transformers) on target datasets
- Feature scaling and G/k selection crucial for nonstationary signals
- Operator learning and surrogate modeling in engineering/physics
- Sectors: mechanics/CFD, materials, climate, digital twins
- Tools/products/workflows:
- ReLU-KAN inside Deep Operator Networks (e.g., in place of KAN layers) to accelerate PDE operator learning and inference
- Surrogate models for design space exploration that train faster and scale better on GPUs
- Assumptions/dependencies:
- PDE/operator benchmarks (Navier–Stokes, elasticity) need validation for accuracy and stability
- Boundary conditions and physics-informed losses may require tailoring of basis coverage (G, k) and regularization
- Model-based control and system identification
- Sectors: robotics, automotive, aerospace, industrial automation
- Tools/products/workflows:
- ReLU-KAN for compact, interpretable local-basis modeling of system dynamics
- Closed-loop pipelines: identify dynamics → MPC/policy optimization → continual updates with minimal forgetting
- Assumptions/dependencies:
- Stability under distribution shift must be verified; consider robust controllers and constraints
- Latency budgets and edge deployment require profiling on target hardware
- Energy-efficient training and “green AI” policy alignment
- Sectors: enterprise AI, public sector, sustainability programs
- Tools/products/workflows:
- Internal guidelines encouraging GPU-parallelizable architectures (ReLU-KAN) to cut training time/energy by 5–20× on relevant tasks
- Procurement/benchmarking policies that account for wall-clock savings for function-approximation workloads
- Assumptions/dependencies:
- Energy savings depend on workload mix and hardware utilization
- Broader tasks (vision/NLP) still need validation to claim systemic savings
- Interpretable local-basis analysis for feature auditing
- Sectors: healthcare, finance, risk/compliance, industrial diagnostics
- Tools/products/workflows:
- Inspect per-dimension basis weights and localized responses to understand which input regions drive predictions
- Diagnostics dashboards for engineers or auditors
- Assumptions/dependencies:
- Interpretability is local (per basis) rather than globally causal; combine with complementary explainability methods
- Regulatory-grade interpretability needs rigorous evaluation and documentation
- Education and training resources
- Sectors: academia, EdTech
- Tools/products/workflows:
- Teaching materials/notebooks demonstrating KAN concepts with compact, fast GPU code
- Labs comparing MLP/KAN/ReLU-KAN for universal approximation and continual learning
- Assumptions/dependencies:
- None beyond standard PyTorch/CUDA setup
Long-Term Applications
These require further research, scaling studies, or engineering to reach production readiness.
- Replacing MLP blocks within large architectures (Transformers/CNNs)
- Sectors: NLP, vision, multimodal, speech
- Tools/products/workflows:
- “ReLU-KAN blocks” as drop-in MLP substitutes in Transformer feed-forward layers for parameter-efficiency or stability
- Hybrid architectures combining attention with localized basis expansions
- Assumptions/dependencies:
- Empirical validation on large-scale datasets is pending
- Training dynamics, initialization, and scaling laws with G/k require study
- Federated and privacy-preserving continual learning
- Sectors: healthcare (EHR/time-series), mobile (keyboard, wearables), finance
- Tools/products/workflows:
- Client-side ReLU-KAN models updated locally with reduced forgetting across tasks/users; server aggregates basis weights
- Differential privacy/secure aggregation leveraging simple op set for efficient secure computation
- Assumptions/dependencies:
- Robustness to heterogeneous client distributions and communication constraints
- Privacy guarantees must be engineered and assessed
- Hardware acceleration and compiler optimizations
- Sectors: semiconductor, edge AI, embedded systems
- Tools/products/workflows:
- Fused kernels or ASIC blocks specialized for ReLU + matmul + elementwise ops; kernel auto-tuning for S/E precomputation patterns
- Quantization-aware training and INT8 deployment (ops are quantization-friendly)
- Assumptions/dependencies:
- Sufficient market pull to justify silicon changes
- Need standardized operator definitions and ONNX/TensorRT primitives
- Interpretable, regulation-friendly AI systems
- Sectors: credit scoring, healthcare decision support, public sector
- Tools/products/workflows:
- Basis-level attribution frameworks tied to input intervals (defined by G and k) for transparent decision regions
- Model governance pipelines documenting local sensitivity and drift across basis functions
- Assumptions/dependencies:
- Proof of compliance-grade interpretability requires audits and external validation
- Risk of misinterpretation if local responses are treated as causal
- Scientific discovery and high-fidelity surrogates
- Sectors: climate modeling, materials design, drug discovery
- Tools/products/workflows:
- Multi-resolution ReLU-KANs (adaptive G/k) for capturing localized phenomena in complex physical systems
- Coupling with physics-informed losses and uncertainty quantification
- Assumptions/dependencies:
- Benchmarks on complex, high-dimensional PDEs and stochastic systems
- Methods for adaptive grid selection and principled regularization
- Lifelong learning in autonomous systems
- Sectors: autonomous vehicles, drones, service robots
- Tools/products/workflows:
- Task-agnostic continual learning where localized bases mitigate interference among skills
- Safety cases integrating monitors for out-of-distribution detection and rollback strategies
- Assumptions/dependencies:
- Extensive real-world testing for safety and reliability
- Tooling for task segmentation and curriculum design
- AutoML search spaces and model toolkits featuring ReLU-KAN
- Sectors: MLOps, platform providers
- Tools/products/workflows:
- AutoML components that search over G, k, layer widths, and placements of ReLU-KAN blocks
- Visualization tools for basis activation maps and pruning/merging of bases
- Assumptions/dependencies:
- Need meta-datasets and search heuristics to control combinatorial complexity
- Integration with existing pipelines (Ray/Tuner, PyTorch Lightning)
- On-device and MCU-class deployment via optimized kernels
- Sectors: wearables, smart home, industrial sensors
- Tools/products/workflows:
- Highly optimized matmul/ReLU kernels or code generation (e.g., TVM) tailoring ReLU-KAN for tiny devices
- Personalized models updated on-device with minimal forgetting
- Assumptions/dependencies:
- Memory footprint for precomputed S/E and basis weights must fit device constraints
- Mixed-precision and sparsity exploitation may be required
Notes on Adoption
- Data dependence: The paper validates on synthetic function families; domain-specific evaluation is essential before high-stakes deployment.
- Hyperparameters: The grid count G and span k control capacity and locality; provide tuning protocols, regularization, and monitoring for overfitting.
- Scalability: Although GPU-friendly, memory scaling with input dimension × (G+k) should be profiled; consider batching and tiling strategies.
- Compatibility: The simple op set supports mainstream frameworks and accelerators (PyTorch, ONNX, TensorRT), enabling straightforward MLOps integration.
- Interpretability: Localized basis functions enable inspection but are not causal explanations; combine with domain knowledge and formal XAI where required.
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