GitHub_collection_pykan/tutorials/Example_10_relativity-addition.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "5d904dee",
   "metadata": {},
   "source": [
    "# Example 10: Relativitistic Velocity Addition"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6465ec94",
   "metadata": {},
   "source": [
    "In this example, we will symbolically regress $f(u,v)=\\frac{u+v}{1+uv}$. In relavitity, we know the rapidity trick $f(u,v)={\\rm tanh}({\\rm arctanh}\\ u+{\\rm arctanh}\\ v)$. Can we rediscover rapidity trick with KAN?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "94056ef6",
   "metadata": {},
   "source": [
    "Intialize model and create dataset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "0a59179d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "cuda\n",
      "checkpoint directory created: ./model\n",
      "saving model version 0.0\n"
     ]
    }
   ],
   "source": [
    "from kan import *\n",
    "\n",
    "device = torch.device('cuda' if torch.cuda.is_available() else 'cpu')\n",
    "print(device)\n",
    "\n",
    "# initialize KAN with G=3\n",
    "model = KAN(width=[2,1,1], grid=10, k=3, device=device)\n",
    "\n",
    "# create dataset\n",
    "f = lambda x: (x[:,[0]]+x[:,[1]])/(1+x[:,[0]]*x[:,[1]])\n",
    "dataset = create_dataset(f, n_var=2, ranges=[-0.9,0.9], device=device)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cb1f817e",
   "metadata": {},
   "source": [
    "Train KAN and plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "a87b97b0",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "| train_loss: 2.28e-03 | test_loss: 2.31e-03 | reg: 6.50e+00 | : 100%|█| 20/20 [00:03<00:00,  5.88it"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "saving model version 0.1\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\n"
     ]
    }
   ],
   "source": [
    "model.fit(dataset, opt=\"LBFGS\", steps=20);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "3f1cfc9d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x400 with 6 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "model.plot(beta=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5ca6421a",
   "metadata": {},
   "source": [
    "Retrain the model, the loss remains similar, meaning that the locking does not degrade model behavior, justifying our hypothesis that these two activation functions are the same. Let's now determine what this function is using $\\texttt{suggest_symbolic}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "2ccb7048",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  function  fitting r2    r2 loss  complexity  complexity loss  total loss\n",
      "0  arctanh    0.999992 -15.786788           4                4  -15.786788\n",
      "1      tan    0.999825 -12.397871           3                3  -12.397871\n",
      "2   arccos    0.998852  -9.753944           4                4   -9.753944\n",
      "3   arcsin    0.998852  -9.753944           4                4   -9.753944\n",
      "4     sqrt    0.982166  -5.808383           2                2   -5.808383\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "('arctanh',\n",
       " (<function kan.utils.<lambda>(x)>,\n",
       "  <function kan.utils.<lambda>(x)>,\n",
       "  4,\n",
       "  <function kan.utils.<lambda>(x, y_th)>),\n",
       " 0.999992311000824,\n",
       " 4)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model.suggest_symbolic(0,1,0,weight_simple=0.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0092be41",
   "metadata": {},
   "source": [
    "We can see that ${\\rm arctanh}$ is at the top of the suggestion list! So we can set both to arctanh, retrain the model, and plot it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "1bb96fe1",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "r2 is 0.9999759197235107\n",
      "saving model version 0.2\n",
      "r2 is 0.999992311000824\n",
      "saving model version 0.3\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "tensor(1.0000, device='cuda:0')"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model.fix_symbolic(0,0,0,'arctanh')\n",
    "model.fix_symbolic(0,1,0,'arctanh')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "83b852a3",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "| train_loss: 7.94e-04 | test_loss: 9.43e-04 | reg: 4.12e+00 | : 100%|█| 20/20 [00:04<00:00,  4.34it"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "saving model version 0.4\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\n"
     ]
    }
   ],
   "source": [
    "model.fit(dataset, opt=\"LBFGS\", steps=20, update_grid=False);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "9ccd0923",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x400 with 6 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "model.plot(beta=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4b98a727",
   "metadata": {},
   "source": [
    "We will see that ${\\rm tanh}$ is at the top of the suggestion list! So we can set it to ${\\rm tanh}$, retrain the model to machine precision, plot it and finally get the symbolic formula."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "99ad38b9",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   function  fitting r2    r2 loss  complexity  complexity loss  total loss\n",
      "0      tanh    0.999998 -16.336284           3                3  -16.336284\n",
      "1    arctan    0.999435 -10.764618           4                4  -10.764618\n",
      "2       cos    0.995899  -7.926177           2                2   -7.926177\n",
      "3       sin    0.995899  -7.926177           2                2   -7.926177\n",
      "4  gaussian    0.994457  -7.492519           3                3   -7.492519\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "('tanh',\n",
       " (<function kan.utils.<lambda>(x)>,\n",
       "  <function kan.utils.<lambda>(x)>,\n",
       "  3,\n",
       "  <function kan.utils.<lambda>(x, y_th)>),\n",
       " 0.9999979138374329,\n",
       " 3)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model.suggest_symbolic(1,0,0,weight_simple=0.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "af24c80d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "r2 is 0.9999979138374329\n",
      "saving model version 0.5\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "tensor(1.0000, device='cuda:0')"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model.fix_symbolic(1,0,0,'tanh')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "01936f17",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "| train_loss: 1.97e-06 | test_loss: 2.06e-06 | reg: 0.00e+00 | : 100%|█| 2000/2000 [00:21<00:00, 93.\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "saving model version 0.6\n"
     ]
    }
   ],
   "source": [
    "model.fit(dataset, opt=\"Adam\", lr=1e-3, steps=2000, update_grid=False, singularity_avoiding=True);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "76bcc188",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x400 with 6 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "model.plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "b62b0246",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\tanh{\\left(\\operatorname{atanh}{\\left(x_{1} \\right)} + \\operatorname{atanh}{\\left(x_{2} \\right)} \\right)}$"
      ],
      "text/plain": [
       "tanh(atanh(x_1) + atanh(x_2))"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "formula = model.symbolic_formula()[0][0]\n",
    "nsimplify(ex_round(formula, 4))"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.16"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}