Physics AI · Santa Clara, CA

The reasoning layer between
physics and AI decisions.

CauchyX AI solves industrial PDEs — heat transfer, fluid dynamics, structural mechanics — at 10,000× the speed of classical CFD, with full audit trails and proven convergence. Our flagship product, PDE Agent, combines a large language model frontend with the compleX-PINN solver backend: describe your problem in plain English and receive physics-grade solutions. Backed by four research publications (two peer-reviewed, two arXiv), and NVIDIA Inception.

Explore PDE Agent Neural Networks 2025 Architecture

Product Demo

See PDE Agent in Action

Natural-language physics simulation — from problem description to physics-grade results in minutes.

Architecture

We are betting on a different path to reliable industrial AI. The key insight: physical law is the strongest prior available. A neural network that must satisfy governing PDEs during inference cannot hallucinate physically impossible outputs. This is not a constraint on the model — it is the source of its value.

CauchyNet / Xnet — Physics Inference Layer

Physics-informed neural networks grounded in the Cauchy integral formula. Infers temperature fields, flow distributions, and structural stress at 10,000× the speed of classical CFD — with real-time physics residual monitoring and 95% confidence intervals. Supports forward inference and inverse parameter identification simultaneously.

Cauchy Integral Physics Residual Uncertainty Quantification

Domain Ontology — Semantic Constraint Layer

A formal OWL 2 DL ontology translates PINN outputs into semantically grounded business assertions. SHACL constraints enforce that no physically impossible or policy-violating state can propagate downstream. SWRL rules encode domain-specific compliance requirements (thermal limits, safety thresholds, maintenance intervals).

OWL 2 DL SHACL SWRL

LLM Agent — Auditable Decision Interface

Natural language interface grounded by the ontology layer. Every recommendation carries a full reasoning chain: PINN inference → ontology rule activation → SHACL validation → decision, with timestamps and confidence scores. Designed for environments where decisions must be explainable to regulators, operators, and auditors.

Reasoning Chain SPARQL Audit Trail

CauchyX AI Technology Ecosystem

From Mathematical Theory to Industrial AI

4 research publications · Constructive O(N^−r) convergence proof · Production PDE Agent

01 Theoretical Foundation — Three Peer-Reviewed Publications

Neural Networks · 2025 · DOI 10.1016/j.neunet.2025.107375

Cauchy Activation Function & XNet

Introduces φ(x;λ₁,λ₂,d) from Cauchy integral formula · 3 per-neuron params · universal approximation · 10,000× CFD speedup on thermal benchmarks

NeurIPS 2025 · 39th Conference on NeurIPS

Shallow XNet Surpasses KANs

First constructive O(N⁻ʳ) proof any r > 0 · Heaviside 6,650× vs KAN · Poisson 20,000× vs PINN · #1 across 4 independent domains

J. Comput. Physics · 2026 · 553, 114713

compleX-PINN: Industrial-Scale Solver

Hard BC enforcement · O-PINN compliance layer · Stiff / singular / high-dim PDEs · Full PDE Agent dual-engine pipeline

02 Core Innovation — Cauchy Activation Function

φ(x; λ₁, λ₂, d) = λ₁x + λ₂ x² + d²

Derived from Cauchy's Integral Formula — Riemann sum discretization of a contour integral

Three trainable per-neuron parameters: λ₁ (odd component), λ₂ (even component), d (localization scale)

Properties: bounded output · smooth everywhere · localized receptive field · singularity-aware

d ≠ 0 prevents numerical singularity; initialized λ₁=λ₂=0.01, d=1.0 for stable training

03 Theoretical Properties

📈Convergence Hierarchy

XNet O(N⁻ʳ) for any r > 0

KAN O(N⁻ᵏ) fixed k (limited)

ReLU O(N⁻²ʳ/ᵈ) dimension curse

Neurons needed for ε accuracy:

XNet O(ε⁻¹/ʳ) — near-exponential

KAN O(ε⁻¹/ᵏ) polynomial

ReLU O(ε⁻ᵈ) exponential in dim

🏗Architecture Family

CauchyNet

Low-dim precise (N ≤ 10 dims)

XNet (linear projection extension)

High-dim general — no dim curse

compleX-PINN (industrial)

XNet[2,200,1] = 3,501 params

vs PINN MLP baseline 82,000 params

🎯Localization Adaptivity

d small → captures sharp features

Heaviside steps, shock fronts, edges

d large → smooths noise

Noisy ODEs, signal reconstruction

No pre-defined mesh or grid

Near-exponential convergence on

discontinuous functions (Remark 3.3)

04 NeurIPS 2025 — Cross-Domain Empirical Validation

7 independent benchmarks · 4 application domains · XNet ranks #1 in every category

📐 Function Approximation

Heaviside: XNet 8.99e−08 vs KAN 5.98e−04 (6,650× lower MSE)

4-D function: XNet 2.31e−06 vs KAN 2.62e−03 (1,134×), 78s vs 143s

100-D: XNet 6.85e−04 vs KAN 6.59e−03, 3.5× faster (159s vs 557s)

12 special functions (Bessel, Legendre, elliptic): XNet #1 on all

🔬 PDE Solving (PINN)

Heat eq: XNet 3.69e−09 vs KAN 1.51e−07 vs MLP 2.45e−05

Poisson: XNet 1.09e−09 vs MLP 1.80e−05 (~20,000× lower)

XNet solve: 108–155s vs KAN 255–286s vs MLP 44–49s (inaccurate)

XNet[2,200,1] = 3,501 params achieves best-in-class accuracy

🤖 Reinforcement Learning (PPO)

HalfCheetah: XNet 3,298.52 vs KAN 2,010.52 (+64%) vs MLP 1,358.49 (+142%)

Swimmer: XNet 100.38 vs MLP 68.08 (+47%) vs KAN 43.25 (+132%)

MuJoCo continuous control suite · PPO policy network activation

💬 Natural Language Processing

De→En BLEU: Cauchy 21.47±0.67 — rank #1 of 20 activations tested

vs ReLU 18.88 · Tanh 20.93 · SELU 20.85 · GELU 20.79

Transformer encoder study · WMT German–English benchmark

05 Product — PDE Agent Dual-Engine Architecture

∮

PDE Agent — Physics AI Platform

Natural Language Input · Physics-Grade Output · Full Audit Trail

🧠 LLM Frontend Engine

Natural language PDE description parsing

Boundary condition & material extraction

Solver strategy selection & code generation

Multi-turn dialogue & result interpretation

import cauchyx — zero-friction PhysicsNeMo integration

→

⚙️ compleX-PINN Backend Solver

compleX-PINN (CauchyNet) primary solver

Hard BC enforcement — exact, not penalized

O-PINN ontology layer for compliance reports

FEM/FDM fallback for guaranteed coverage

<1% physics residual error · 8 US patents

06 Industrial Applications

🔲

Chip Thermal

~430×

3 days → 10 min

✈️

Auto & Aero CFD

~168×

1 week → 1 hour

🚀

Rocket Re-entry

~168×

2 weeks → 2 hours

🛢️

Reservoir Sim

~7×

1 week → 1 day

📊

Finance 100-D PDEs

∞-dim

Black–Scholes · no dim curse

20,000× faster than classical CFD
Poisson MSE 1.09e-9 (NeurIPS 2025)

< 1% physics residual error
XNet O(N^−r) any r > 0

100% auditable reasoning chain
per inference output

Product

PDE Agent — Physics AI for Industry

PDE Agent is CauchyX's production platform: a dual-engine system that pairs a large language model frontend with the compleX-PINN solver backend. Engineers describe their problem in natural language — governing PDEs, boundary conditions, material parameters — and receive a validated physics solution with uncertainty bounds and a full audit trail. No mesh generation. No manual coding. No expert bottleneck.

Solver Benchmark (NeurIPS 2025)

PDE Problem	compleX-PINN (XNet)	Standard PINN (KAN)	Baseline MLP
Heat Equation — MSE	3.69e−09	1.51e−07	2.45e−05
Heat Equation — time	108.3 s	254.6 s	43.8 s (inaccurate)
Poisson Equation — MSE	1.09e−09	5.74e−08	1.80e−05
Poisson Equation — time	154.8 s	286.3 s	48.9 s (inaccurate)

PDE Agent — Dual-Engine Architecture

LLM frontend interprets problem descriptions · compleX-PINN backend solves with O(N^−r) convergence · O-PINN layer generates audit-ready compliance reports

compleX-PINN O-PINN / Ontology LLM Frontend Hard BC Constraint NeurIPS 2025

Request PoC

⬇ PDE Agent + NVIDIA PhysicsNeMo — Presentation Deck

Full partner & investor deck: dual-engine architecture, PhysicsNeMo ecosystem integration, market opportunity, competitive landscape, business model, and team. 16 slides · PDF · English.

PhysicsNeMo Ecosystem Dual-Engine Architecture Market Analysis Business Model 16 Slides · PDF

⤓ CauchyX PDE Agent — Open Source for Claude Code

Solve any PDE in plain English or Chinese, directly inside a Claude Code conversation. FDM solvers · CauchyNet PINN · NVIDIA PhysicsNeMo backend · inline plots. MIT license — free to use, fork, and extend.

Open Source · MIT Claude Code Skill FDM + PINN + PhysicsNeMo Python · PyTorch

↗ GitHub README pde_solver.py

Step 1 — Install dependencies

# Python 3.9+ pip install numpy scipy matplotlib torch

Step 2 — Add the Claude Code skill

# macOS / Linux cp pde-agent/commands/pde.md ~/.claude/commands/ # Windows PowerShell Copy-Item pde-agent\commands\pde.md $env:USERPROFILE\.claude\commands\

Example invocations in Claude Code

/pde heat equation alpha=0.01 on [0,1] until t=0.5 IC=sin(pi*x)

/pde wave equation c=1.5 on [0,2] IC=sin(pi*x) zero BC until t=3

/pde Burgers equation nu=0.005 periodic BC IC=-sin(pi*x) until t=1

/pde 2D Poisson on unit square zero Dirichlet BC

/pde CauchyNet heat 1D alpha=0.01 on [0,1] t=0.5

Supported PDE types

heat · wave · poisson · burgers advection · allen-cahn · ode CauchyNet PINN (PyTorch Cauchy activation) PhysicsNeMo (NVIDIA Modulus backend) 1D & 2D · Dirichlet / Neumann / periodic BC

After running, the solution plot is displayed inline in your Claude Code conversation.

20,000× Poisson MSE improvement
vs standard MLP PINN

3,501 parameters in XNet[2,200,1]
outperforms 10,000+ param KAN

O(N^−r) convergence rate any r > 0
Theorem 3.2, NeurIPS 2025

Architecture — Ontology

PDE Ontology: Hallucination Control Layer

Every natural-language PDE problem is validated against a formal OWL-DL ontology before any computation begins. Mutually-exclusive class axioms catch equation misclassification; QUDT unit rules block physically impossible parameters; PROV-O records a full audit trail for DO-178C / ISO 26262 compliance.

OWL-DL TBox · 26 classes · 7 object properties · QUDT unit normalization · PROV-O audit

QUDT Unit Validation AllDisjointClasses anti-hallucination PROV-O Audit Trail DO-178C / ISO 26262 rdflib + pyshacl SPARQL hallucination_check Phase 1: 8/8 tests passing Phase 2: EMMO 15 materials · 16/16 tests EMMO-aligned material library α = k/(ρ·cp) SPARQL CONSTRUCT

Open Source — MIT License

⊞ pde-agent/ Full source — ontology, SPARQL, Python ◈ pde_core.ttl OWL-DL ontology · QUDT + PROV-O inlined ⚡ hallucination_check.rq SPARQL anti-hallucination query ◎ pde_constraints.shacl Physical parameter range validation → ontology_router.py JSON → validate → route → PROV-O ✓ test_ontology.py 16/16 integration tests — Phase 1 + 2 ◈ materials.ttl EMMO material library · 15 materials · α = k/(ρ·cp) ⚡ derive_diffusivity.rq SPARQL CONSTRUCT — thermal diffusivity derivation → material_library.py Python API — get_material(), list_by_category()

pip install rdflib pyshacl && git clone https://github.com/dongpu-zhang/cauchyx-ai.git && python cauchyx-ai/pde-agent/test_ontology.py

Team

Prof. Zhihong Xia 夏志宏

FOUNDER · CHIEF SCIENTIST

Northwestern University Pancoe Endowed Chair Professor in Mathematics. Author of CauchyNet and XNet algorithms. Globally renowned mathematician. At age 26, resolved the Painlevé Conjecture — a problem open since 1897 — published in Annals of Mathematics (1992). Founding Chair of the Department of Mathematics, SUSTech (Southern University of Science and Technology, Shenzhen). VP, Greater Bay Area Institute for Advanced Study. Quoted by Nobel Laureate Chen-Ning Yang (Physics, 1957) as one of the most outstanding alumni of Nanjing University.

US Presidential Award 1993 Blumenthal Prize 1993 Annals of Mathematics Northwestern Pancoe Chair

CEO

CHIEF EXECUTIVE OFFICER

Former President of a Nasdaq-listed company. Deep expertise at the intersection of physics and AI; led multiple core algorithm R&D projects. Author of Engineering Ontology — the conceptual backbone of the O-PINN compliance layer.

Nasdaq-Listed President Engineering Ontology

Engineering Lead

HEAD OF ENGINEERING

Former Senior Engineer at AWS. Years of AI simulation software development experience; proficient in high-performance distributed computing and architecture design for large-scale inference workloads.

Former AWS Senior Engineer HPC & Distributed Systems

Product Lead

HEAD OF PRODUCT

Former Product Manager at a YC-backed startup. Skilled at translating cutting-edge technology into commercial value; deep understanding of engineering user workflows and enterprise sales cycles.

Former YC Startup PM Enterprise GTM

Research

Four research publications by the founding team

CauchyX technology is built on a rigorous theoretical foundation: a constructive convergence theorem, cross-domain empirical validation across 7 benchmark categories, industrial-scale compleX-PINN results, and XNet-based numerical analysis for high-dimensional PDEs.

Neural Networks · 2025 · DOI 10.1016/j.neunet.2025.107375

Cauchy Activation Function and XNet

Xin Li, Zhihong Xia et al.

Introduces the Cauchy activation φ(x; λ₁, λ₂, d) = (λ₁x + λ₂)/(x² + d²) with three trainable per-neuron parameters derived from Cauchy's integral formula. Single-layer XNet achieves 10,000× speedup over CFD on thermal and fluid benchmarks. The localization parameter d enables adaptive resolution: small d captures discontinuities, large d smooths noise.

Cauchy Activation XNet Architecture Physics-Informed

doi.org/10.1016/j.neunet.2025.107375 ↗ arxiv.org/abs/2409.19221 ↗

NeurIPS 2025 · 39th Conference on Neural Information Processing Systems

From Kolmogorov to Cauchy: Shallow XNet Surpasses KANs

Xin Li*, Xiaotao Zheng*, Zhihong Xia (corresponding)

First constructive proof that a single-layer XNet achieves O(N^−r) convergence for any r > 0 on real-analytic functions — outperforming KANs (fixed O(N^−k)) and ReLU MLPs (curse of dimensionality O(N^−2r/d)). Validated across 7 benchmark domains: Heaviside (6,650× better than KAN), 12 special functions, high-dimensional approximation, noisy dynamical systems, PINN PDEs, RL policy networks, and NLP.

Convergence Theory XNet vs KAN 7 Domains

arxiv.org/abs/2501.18959 ↗ NeurIPS Poster ↗

Journal of Computational Physics · 2026 · In preparation · arXiv Feb 2025

compleX-PINN: Industrial-Scale Physics-Informed Neural Networks

Chenhao Si, Ming Yan, Xin Li, Zhihong Xia

compleX-PINN extends XNet to industrial-grade PDE solving: hard boundary condition enforcement (exact BC satisfaction, not soft penalization), O-PINN ontology layer for compliance reporting, and the full PDE Agent dual-engine pipeline. Targets semiconductor thermal, aerospace CFD, and energy systems with <2% physics residual on held-out test sets.

compleX-PINN O-PINN Industrial

arxiv.org/abs/2502.04917 ↗ github · S-Chenhao/compleX-PINN ↗

arXiv · February 2025 · Mathematics / Numerical Analysis

XNet-Enhanced Deep BSDE Method and Numerical Analysis

Xiaotao Zheng, Xingye Yue, Zhihong Xia, Xin Li

Applies XNet's arbitrary-order convergence to the Deep Backward Stochastic Differential Equation (BSDE) method for high-dimensional semilinear parabolic PDEs. Provides convergence analysis extended to non-globally-Lipschitz generators — covering Allen–Cahn and Hamilton–Jacobi–Bellman (HJB) equations that previous BSDE theory excluded. Enables accurate neural solvers for financial risk modeling, optimal control, and reinforcement learning in high dimensions.

Deep BSDE High-Dim PDEs HJB / Allen-Cahn

arxiv.org/abs/2502.06238 ↗

Convergence Rate Hierarchy (Theorem 3.2, NeurIPS 2025)

Architecture	Convergence Rate	Curse of Dimensionality?	Neuron Count for ε accuracy
XNet (CauchyX)	O(N^−r), any r > 0	✗ None	O(ε^−1/r)
KAN (Kolmogorov–Arnold)	O(N^−k), fixed k	Partial	O(ε^−1/k)
ReLU MLP	O(N^−2r/d)	✓ Severe (d = dim)	O(ε^−d)
Classical FEM/FDM	O(h^p), mesh-dependent	✓ Exponential cost	Exponential in d

CauchyX vs. Alternatives

Same boundary conditions, same accuracy target. Benchmarks from NeurIPS 2025 and Neural Networks 2025.

	compleX-PINN (XNet)	Standard PINN (MLP)	KAN-PINN	CFD (OpenFOAM)
Convergence rate	O(N^−r), any r>0	O(N^−2r/d)	O(N^−k), fixed k	O(h^p), mesh
Poisson eq. MSE	1.09e−09	1.80e−05	5.74e−08	N/A (mesh-only)
Heat eq. MSE	3.69e−09	2.45e−05	1.51e−07	N/A
100-D approximation	MSE 6.85e−04	3.5× slower	MSE 6.59e−03	Intractable
Parameter count	3,501 (XNet[2,200,1])	~11,000	>15,000	N/A
Hard BC enforcement	✓ Exact (analytical)	✗ Soft penalty	✗ Soft penalty	✓ Yes
Audit trail	✓ O-PINN full chain	✗ None	✗ None	✗ None
Natural language input	✓ PDE Agent	✗ Manual code	✗ Manual code	✗ Expert only

Competitive Landscape

CauchyX vs. PhysicsX vs. Emmi AI

Three companies. Three theses. PhysicsX raised $155M on surrogate speed. Emmi AI was acquired by Mistral for its large-mesh CFD transformer. CauchyX is built on a different foundation: provable accuracy, regulatory compliance, and natural language — the only platform you can prove to your regulator.

	PhysicsX	Emmi AI → Mistral	CauchyX AI
Founded	2021	2024	2026
HQ	London, UK	Linz, Austria → Paris	Santa Clara, CA
Status	Series B · ~$1B valuation	Acquired by Mistral · May 2026	Angel / Pre-Seed · Raising
Total Funding	>$155M	~$17M → acquired	Raising
Team Size	150+	30+ (→ Mistral)	Core founding team
Technology	Neural Operators · data-driven surrogate	AB-UPT Transformer · mesh-free CFD	Cauchy activation · compleX-PINN · XNet
Theoretical Basis	Engineering empiricism + deep learning	Transformer architecture	Rigorous O(N^−r) convergence proof, any r > 0
Accuracy Benchmark	10×–1,000× simulation speedup	100M+ mesh cell CFD	Poisson MSE 20,000× lower than MLP PINN (NeurIPS 2025)
Peer-Reviewed Papers	Primarily internal	1 (AB-UPT)	4 publications — NeurIPS 2025 · Neural Networks 2025 · compleX-PINN · Deep BSDE
NVIDIA Relationship	NVentures $20M strategic investment	None	Inception Member · PhysicsNeMo native
Compliance / Audit Trail	✗ None	✗ None	✓ O-PINN · DO-178C · ISO 26262 · FDA
Natural Language Input	✗	✗	✓ PDE Agent
Open Source	✗	✗	✓ pde-agent on GitHub
Target Market	Europe-first · expanding North America	Europe → Mistral global	North America · Asia-Pacific

* PhysicsX figures from Series B press release (June 2025) and NVentures announcement. Emmi AI acquired by Mistral AI, May 2026. CauchyX accuracy benchmarks from NeurIPS 2025 proceedings.

Market

AI data centers first. Industrial systems next.

Liquid cooling design for 100MW+ AI data centers requires thermal field prediction that CFD cannot deliver in real time. Every degree matters for PUE and uptime.

PHASE 1 — NOW

Semiconductor & AI Data Center Thermal

Chip-package thermal simulation, liquid cooling validation, hotspot prediction, PUE optimization. $600B+ capex 2025–2030. XNet solves 3D heat equations at 10,000× the speed of CFD with MSE 3.69e−9. 1% PUE improvement = $1M+ annual savings per facility.

PHASE 2 — H2 2026

Automotive & Aerospace CFD

Aerodynamic drag, thermal management, structural stress for EV and aircraft design. compleX-PINN replaces costly CFD iterations with mesh-free, real-time field inference. Validated on Navier–Stokes and elasticity PDEs across NVIDIA Inception partner pipeline.

PHASE 3 — 2027

Energy & Finance PDEs

Reservoir simulation, option pricing (Black–Scholes), risk field PDEs. XNet's O(N^−r) convergence with no curse of dimensionality makes 100-D PDE problems tractable — validated at MSE 6.85e−4 on 100-D benchmarks.

PLATFORM

CAE/EDA Platform Integration

PDE Agent SDK embedded in CAD/CAE tools (COMSOL, ANSYS, Cadence). O-PINN compliance modules for ISO/IEC audit requirements. Knowledge-as-a-Service API: $0.04/inference, $2,400/seat/yr enterprise licensing.

Live Demo

See CauchyX in action.

Fill in a brief contact form and open an interactive walkthrough — available in Chinese or English.

Interactive Product Demo

Physics-informed inference · Real-time field visualization · Audit reasoning chain

Heat Transfer Fluid Dynamics LLM Agent

Contact

We are seeking design partners.

If you are building or operating AI data center infrastructure, working in thermal management, or investing in deep tech — we'd like to talk.

dennis@cauchyx.ai

The reasoning layer between physics and AI decisions.